Mapping types use the syntax mapping(KeyType KeyName? => ValueType ValueName?) and variables of mapping type are declared using the syntax mapping(KeyType KeyName? => ValueType ValueName?) VariableName.
The KeyType can be any built-in value type, bytes, string, or any contract or enum type.
Other user-defined or complex types, such as mappings, structs or array types are not allowed.
ValueType can be any type, including mappings, arrays and structs.
KeyName and ValueName are optional (so mapping(KeyType => ValueType) works as well) and can be any valid identifier that is not a type.

You can think of mappings as hash tables, which are virtually initialised such that every possible key exists and is mapped to a value whose byte-representation is all zeros, a type’s default value.
The similarity ends there, the key data is not stored in a mapping, only its keccak256 hash is used to look up the value.

Because of this, mappings do not have a length or a concept of a key or value being set, and therefore cannot be erased without extra information regarding the assigned keys (see Clearing Mappings).

Mappings can only have a data location of storage and thus are allowed for state variables, as storage reference types in functions, or as parameters for library functions.
They cannot be used as parameters or return parameters of contract functions that are publicly visible.
These restrictions are also true for arrays and structs that contain mappings.

You can mark state variables of mapping type as public and Solidity creates a getter for you.
The KeyType becomes a parameter with name KeyName (if specified) for the getter.
If ValueType is a value type or a struct, the getter returns ValueType with name ValueName (if specified).
If ValueType is an array or a mapping, the getter has one parameter for each KeyType, recursively.

In the example below, the MappingExample contract defines a public balances mapping, with the key type an address, and a value type a uint, mapping an Ethereum address to an unsigned integer value.
As uint is a value type, the getter returns a value that matches the type, which you can see in the MappingUser contract that returns the value at the specified address.

// SPDX-License-Identifier: GPL-3.0
pragma solidity >=0.4.0 <0.9.0;

contract MappingExample {
    mapping(address => uint) public balances;

    function update(uint newBalance) public {
        balances[msg.sender] = newBalance;
    }
}

contract MappingUser {
    function f() public returns (uint) {
        MappingExample m = new MappingExample();
        m.update(100);
        return m.balances(address(this));
    }
}

The example below is a simplified version of an ERC20 token.
_allowances is an example of a mapping type inside another mapping type.

In the example below, the optional KeyName and ValueName are provided for the mapping.
It does not affect any contract functionality or bytecode, it only sets the name field for the inputs and outputs in the ABI for the mapping’s getter.

// SPDX-License-Identifier: GPL-3.0
pragma solidity ^0.8.18;

contract MappingExampleWithNames {
    mapping(address user => uint balance) public balances;

    function update(uint newBalance) public {
        balances[msg.sender] = newBalance;
    }
}

The example below uses _allowances to record the amount someone else is allowed to withdraw from your account.

// SPDX-License-Identifier: GPL-3.0
pragma solidity >=0.4.22 <0.9.0;

contract MappingExample {

    mapping (address => uint256) private _balances;
    mapping (address => mapping (address => uint256)) private _allowances;

    event Transfer(address indexed from, address indexed to, uint256 value);
    event Approval(address indexed owner, address indexed spender, uint256 value);

    function allowance(address owner, address spender) public view returns (uint256) {
        return _allowances[owner][spender];
    }

    function transferFrom(address sender, address recipient, uint256 amount) public returns (bool) {
        require(_allowances[sender][msg.sender] >= amount, "ERC20: Allowance not high enough.");
        _allowances[sender][msg.sender] -= amount;
        _transfer(sender, recipient, amount);
        return true;
    }

    function approve(address spender, uint256 amount) public returns (bool) {
        require(spender != address(0), "ERC20: approve to the zero address");

        _allowances[msg.sender][spender] = amount;
        emit Approval(msg.sender, spender, amount);
        return true;
    }

    function _transfer(address sender, address recipient, uint256 amount) internal {
        require(sender != address(0), "ERC20: transfer from the zero address");
        require(recipient != address(0), "ERC20: transfer to the zero address");
        require(_balances[sender] >= amount, "ERC20: Not enough funds.");

        _balances[sender] -= amount;
        _balances[recipient] += amount;
        emit Transfer(sender, recipient, amount);
    }
}

You cannot iterate over mappings, i.e. you cannot enumerate their keys.
It is possible, though, to implement a data structure on top of them and iterate over that.
For example, the code below implements an IterableMapping library that the User contract then adds data to, and the sum function iterates over to sum all the values.

// SPDX-License-Identifier: GPL-3.0
pragma solidity ^0.8.8;

struct IndexValue { uint keyIndex; uint value; }
struct KeyFlag { uint key; bool deleted; }

struct itmap {
    mapping(uint => IndexValue) data;
    KeyFlag[] keys;
    uint size;
}

type Iterator is uint;

library IterableMapping {
    function insert(itmap storage self, uint key, uint value) internal returns (bool replaced) {
        uint keyIndex = self.data[key].keyIndex;
        self.data[key].value = value;
        if (keyIndex > 0)
            return true;
        else {
            keyIndex = self.keys.length;
            self.keys.push();
            self.data[key].keyIndex = keyIndex + 1;
            self.keys[keyIndex].key = key;
            self.size++;
            return false;
        }
    }

    function remove(itmap storage self, uint key) internal returns (bool success) {
        uint keyIndex = self.data[key].keyIndex;
        if (keyIndex == 0)
            return false;
        delete self.data[key];
        self.keys[keyIndex - 1].deleted = true;
        self.size --;
    }

    function contains(itmap storage self, uint key) internal view returns (bool) {
        return self.data[key].keyIndex > 0;
    }

    function iterateStart(itmap storage self) internal view returns (Iterator) {
        return iteratorSkipDeleted(self, 0);
    }

    function iterateValid(itmap storage self, Iterator iterator) internal view returns (bool) {
        return Iterator.unwrap(iterator) < self.keys.length;
    }

    function iterateNext(itmap storage self, Iterator iterator) internal view returns (Iterator) {
        return iteratorSkipDeleted(self, Iterator.unwrap(iterator) + 1);
    }

    function iterateGet(itmap storage self, Iterator iterator) internal view returns (uint key, uint value) {
        uint keyIndex = Iterator.unwrap(iterator);
        key = self.keys[keyIndex].key;
        value = self.data[key].value;
    }

    function iteratorSkipDeleted(itmap storage self, uint keyIndex) private view returns (Iterator) {
        while (keyIndex < self.keys.length && self.keys[keyIndex].deleted)
            keyIndex++;
        return Iterator.wrap(keyIndex);
    }
}

// How to use it
contract User {
    // Just a struct holding our data.
    itmap data;
    // Apply library functions to the data type.
    using IterableMapping for itmap;

    // Insert something
    function insert(uint k, uint v) public returns (uint size) {
        // This calls IterableMapping.insert(data, k, v)
        data.insert(k, v);
        // We can still access members of the struct,
        // but we should take care not to mess with them.
        return data.size;
    }

    // Computes the sum of all stored data.
    function sum() public view returns (uint s) {
        for (
            Iterator i = data.iterateStart();
            data.iterateValid(i);
            i = data.iterateNext(i)
        ) {
            (, uint value) = data.iterateGet(i);
            s += value;
        }
    }
}

Arithmetic and bit operators can be applied even if the two operands do not have the same type.
For example, you can compute y = x + z, where x is a uint8 and z has the type uint32.
In these cases, the following mechanism will be used to determine the type in which the operation is computed (this is important in case of overflow) and the type of the operator’s result:

1. If the type of the right operand can be implicitly converted to the type of the left operand, use the type of the left operand,
2. if the type of the left operand can be implicitly converted to the type of the right operand, use the type of the right operand,
3. otherwise, the operation is not allowed.

In case one of the operands is a literal number it is first converted to its “mobile type”, which is the smallest type that can hold the value (unsigned types of the same bit-width are considered “smaller” than the signed types).
If both are literal numbers, the operation is computed with effectively unlimited precision in that the expression is evaluated to whatever precision is necessary so that none is lost when the result is used with a non-literal type.

The operator’s result type is the same as the type the operation is performed in, except for comparison operators where the result is always bool.

The operators ** (exponentiation), << and >> use the type of the left operand for the operation and the result.

The ternary operator is used in expressions of the form <expression> ? <trueExpression> : <falseExpression>.
It evaluates one of the latter two given expressions depending upon the result of the evaluation of the main <expression>.
If <expression> evaluates to true, then <trueExpression> will be evaluated, otherwise <falseExpression> is evaluated.

The result of the ternary operator does not have a rational number type, even if all of its operands are rational number literals.
The result type is determined from the types of the two operands in the same way as above, converting to their mobile type first if required.

As a consequence, 255 + (true ? 1 : 0) will revert due to arithmetic overflow.
The reason is that (true ? 1 : 0) is of uint8 type, which forces the addition to be performed in uint8 as well, and 256 exceeds the range allowed for this type.

Another consequence is that an expression like 1.5 + 1.5 is valid but 1.5 + (true ? 1.5 : 2.5) is not.
This is because the former is a rational expression evaluated in unlimited precision and only its final value matters.
The latter involves a conversion of a fractional rational number to an integer, which is currently disallowed.

If a is an LValue (i.e. a variable or something that can be assigned to), the following operators are available as shorthands:

a += e is equivalent to a = a + e.
The operators -=, *=, /=, %=, |=, &=, ^=, <<= and >>= are defined accordingly.
a++ and a-- are equivalent to a += 1 / a -= 1 but the expression itself still has the previous value of a.
In contrast, --a and ++a have the same effect on a but return the value after the change.

delete a assigns the initial value for the type to a.
I.e. for integers it is equivalent to a = 0, but it can also be used on arrays, where it assigns a dynamic array of length zero or a static array of the same length with all elements set to their initial value.
delete a[x] deletes the item at index x of the array and leaves all other elements and the length of the array untouched.
This especially means that it leaves a gap in the array.
If you plan to remove items, a mapping is probably a better choice.

For structs, it assigns a struct with all members reset.
In other words, the value of a after delete a is the same as if a would be declared without assignment, with the following caveat:

delete has no effect on mappings (as the keys of mappings may be arbitrary and are generally unknown).
So if you delete a struct, it will reset all members that are not mappings and also recurse into the members unless they are mappings.
However, individual keys and what they map to can be deleted: If a is a mapping, then delete a[x] will delete the value stored at x.

It is important to note that delete a really behaves like an assignment to a, i.e. it stores a new object in a.
This distinction is visible when a is reference variable: It will only reset a itself, not the value it referred to previously.

// SPDX-License-Identifier: GPL-3.0
pragma solidity >=0.4.0 <0.9.0;

contract DeleteExample {
    uint data;
    uint[] dataArray;

    function f() public {
        uint x = data;
        delete x; // sets x to 0, does not affect data
        delete data; // sets data to 0, does not affect x
        uint[] storage y = dataArray;
        delete dataArray; // this sets dataArray.length to zero, but as uint[] is a complex object, also
        // y is affected which is an alias to the storage object
        // On the other hand: "delete y" is not valid, as assignments to local variables
        // referencing storage objects can only be made from existing storage objects.
        assert(y.length == 0);
    }
}

The following is the order of precedence for operators, listed in order of evaluation.

An implicit type conversion is automatically applied by the compiler in some cases during assignments, when passing arguments to functions and when applying operators.
In general, an implicit conversion between value-types is possible if it makes sense semantically and no information is lost.

For example, uint8 is convertible to uint16 and int128 to int256, but int8 is not convertible to uint256, because uint256 cannot hold values such as -1.

If an operator is applied to different types, the compiler tries to implicitly convert one of the operands to the type of the other (the same is true for assignments).
This means that operations are always performed in the type of one of the operands.

For more details about which implicit conversions are possible, please consult the sections about the types themselves.

In the example below, y and z, the operands of the addition, do not have the same type, but uint8 can be implicitly converted to uint16 and not vice-versa.
Because of that, y is converted to the type of z before the addition is performed in the uint16 type.
The resulting type of the expression y + z is uint16.
Because it is assigned to a variable of type uint32 another implicit conversion is performed after the addition.

uint8 y;
uint16 z;
uint32 x = y + z;

If the compiler does not allow implicit conversion but you are confident a conversion will work, an explicit type conversion is sometimes possible.
This may result in unexpected behaviour and allows you to bypass some security features of the compiler, so be sure to test that the result is what you want and expect!

Take the following example that converts a negative int to a uint:

int  y = -3;
uint x = uint(y);

At the end of this code snippet, x will have the value 0xfffff..fd (64 hex characters), which is -3 in the two’s complement representation of 256 bits.

If an integer is explicitly converted to a smaller type, higher-order bits are cut off:

uint32 a = 0x12345678;
uint16 b = uint16(a); // b will be 0x5678 now

If an integer is explicitly converted to a larger type, it is padded on the left (i.e., at the higher order end).
The result of the conversion will compare equal to the original integer:

uint16 a = 0x1234;
uint32 b = uint32(a); // b will be 0x00001234 now
assert(a == b);

Fixed-size bytes types behave differently during conversions.
They can be thought of as sequences of individual bytes and converting to a smaller type will cut off the sequence:

bytes2 a = 0x1234;
bytes1 b = bytes1(a); // b will be 0x12

If a fixed-size bytes type is explicitly converted to a larger type, it is padded on the right.
Accessing the byte at a fixed index will result in the same value before and after the conversion (if the index is still in range):

bytes2 a = 0x1234;
bytes4 b = bytes4(a); // b will be 0x12340000
assert(a[0] == b[0]);
assert(a[1] == b[1]);

Since integers and fixed-size byte arrays behave differently when truncating or padding, explicit conversions between integers and fixed-size byte arrays are only allowed, if both have the same size.
If you want to convert between integers and fixed-size byte arrays of different size, you have to use intermediate conversions that make the desired truncation and padding rules explicit:

bytes2 a = 0x1234;
uint32 b = uint16(a); // b will be 0x00001234
uint32 c = uint32(bytes4(a)); // c will be 0x12340000
uint8 d = uint8(uint16(a)); // d will be 0x34
uint8 e = uint8(bytes1(a)); // e will be 0x12

bytes arrays and bytes calldata slices can be converted explicitly to fixed bytes types (bytes1/…/bytes32).
In case the array is longer than the target fixed bytes type, truncation at the end will happen. If the array is shorter than the target type, it will be padded with zeros at the end.

// SPDX-License-Identifier: GPL-3.0
pragma solidity ^0.8.5;

contract C {
    bytes s = "abcdefgh";
    function f(bytes calldata c, bytes memory m) public view returns (bytes16, bytes3) {
        require(c.length == 16, "");
        bytes16 b = bytes16(m);  // if length of m is greater than 16, truncation will happen
        b = bytes16(s);  // padded on the right, so result is "abcdefgh\0\0\0\0\0\0\0\0"
        bytes3 b1 = bytes3(s); // truncated, b1 equals to "abc"
        b = bytes16(c[:8]);  // also padded with zeros
        return (b, b1);
    }
}

Decimal and hexadecimal number literals can be implicitly converted to any integer type that is large enough to represent it without truncation:

uint8 a = 12; // fine
uint32 b = 1234; // fine
uint16 c = 0x123456; // fails, since it would have to truncate to 0x3456

Decimal number literals cannot be implicitly converted to fixed-size byte arrays.
Hexadecimal number literals can be, but only if the number of hex digits exactly fits the size of the bytes type.
As an exception both decimal and hexadecimal literals which have a value of zero can be converted to any fixed-size bytes type:

bytes2 a = 54321; // not allowed
bytes2 b = 0x12; // not allowed
bytes2 c = 0x123; // not allowed
bytes2 d = 0x1234; // fine
bytes2 e = 0x0012; // fine
bytes4 f = 0; // fine
bytes4 g = 0x0; // fine

String literals and hex string literals can be implicitly converted to fixed-size byte arrays, if their number of characters matches the size of the bytes type:

bytes2 a = hex"1234"; // fine
bytes2 b = "xy"; // fine
bytes2 c = hex"12"; // not allowed
bytes2 d = hex"123"; // not allowed
bytes2 e = "x"; // not allowed
bytes2 f = "xyz"; // not allowed

As described in Address Literals, hex literals of the correct size that pass the checksum test are of address type.
No other literals can be implicitly converted to the address type.

Explicit conversions to address are allowed only from bytes20 and uint160.

An address a can be converted explicitly to address payable via payable(a).