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Closures are anonymous functions. The full form:

{ (params) -> ReturnType in
    statements
}

Most of the time you can drop the types and the explicit return:

let add: (Int, Int) -> Int = { a, b in a + b }
let double: (Int) -> Int = { $0 * 2 }       // shorthand args

When a closure is the LAST argument, you can use trailing closure syntax — pull the closure out of the parens:

let doubled = [1, 2, 3].map { $0 * 2 }            // trailing
let doubled2 = [1, 2, 3].map({ $0 * 2 })          // also valid

Most of Swift's standard collection methods are designed for trailing-closure syntax — map, filter, sorted, reduce, forEach, etc.