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And later I want to push/append an element to the end of said array, to get:

["Steve", "Bill", "Linus", "Bret", "Tim"]

What method should I use?

And what about the case where I want to add an element to the front of the array? Is there a constant time unshift?

As of Swift 3 / 4 / 5 , this is done as follows.

To add a new element to the end of an Array.

anArray.append("This String")
To append a different Array to the end of your Array.
anArray += ["Moar", "Strings"]
anArray.append(contentsOf: ["Moar", "Strings"])
To insert a new element into your Array.
anArray.insert("This String", at: 0)
To insert the contents of a different Array into your Array.
anArray.insert(contentsOf: ["Moar", "Strings"], at: 0)
More information can be found in the "Collection Types" chapter of "The Swift Programming Language", starting on page 110.
                Great answer. The inconsistency in the way Apple implemented this is kind of irritating. Presumably, anArray+="Foo" does the same thing as anArray+=["Foo"] Personally, I'll only be using the latter syntax to avoid confusing myself.
– original_username
                Jul 7, 2014 at 5:56
                It's failing at run time. There are elements already in the array that I'm attempting to insert, so that's not it. My workaround is to create a new array with the element that I'm attempting to insert, and then append all the elements from the original array. Pretty sure this is the result of a retain problem elsewhere - somehow these objects are fine so long as they're left in their original array spots, but if iOS attempts to move them (as a result of the insert) there's a crash. Or else this is some weird Swift/IB problem.
– MusiGenesis
                Aug 14, 2015 at 13:24
var myArray = ["Steve", "Bill", "Linus", "Bret"]
//add to the end of the array with append
myArray.append(str1)
myArray.append(str2)
To add them to the front:
//use 'insert' instead of append
myArray.insert(str1, atIndex:0)
myArray.insert(str2, atIndex:0)
//Swift 3
myArray.insert(str1, at: 0)
myArray.insert(str2, at: 0)
As others have already stated, you can no longer use '+=' as of xCode 6.1
                Makes sense, with the lack of a .prepend() function, inserting at the head seems to be a good approach
– Atticus
                Oct 24, 2014 at 23:40
That operator is generally equivalent to the append(contentsOf:) method. (And in really old Swift versions, could append single elements, not just other collections of the same element type.)
There's also insert(_:at:) for inserting at any index.
If, say, you'd like a convenience function for inserting at the beginning, you could add it to the Array class with an extension.
                If myArray is of type Array, then the first line is incorrect. The += operator can only be used if both myArray and the right operand are of type Array.
– Bart Jacobs
                Jan 26, 2015 at 15:08
                It was correct back when I posted this answer, but the language has changed since. Answer updated.
– rickster
                Jan 26, 2015 at 15:33
                Saying that += and append are equivalent is not correct, the former involves the creation of a temporary array that will be discarded as soon as the operator function consumes it.
– Cristik
                Feb 25, 2019 at 22:04
                In the second function, there is no need to iterate over the left array. Doing var copy = left and then appending copy.append(right) is enough. In Swift, arrays are value types, not reference types.
– Natan R.
                May 28, 2020 at 12:59
                You might want too add extension Array where Element == YourClass { if it is not compiling
– itMaxence
                Oct 21, 2021 at 16:31
Here is a small extension if you wish to insert at the beginning of the array without loosing the item at the first position
extension Array{
    mutating func appendAtBeginning(newItem : Element){
        let copy = self
        self = []
        self.append(newItem)
        self.appendContentsOf(copy)
let strigTwo = "Two"
let stringThree = "Three"
var array:[String] = []//If your array is string type
Type 1)
//To append elements at the end
array.append(stringOne)
array.append(stringThree)
Type 2)
//To add elements at specific index
array.insert(strigTwo, at: 1)
If you want to add two arrays
var array1 = [1,2,3,4,5]
let array2 = [6,7,8,9]
let array3 = array1+array2
print(array3)
array1.append(contentsOf: array2)
print(array1)
Use Deque instead of Array
The main benefit of Deque over Array is that it supports efficient insertions and removals at both ends.
https://swift.org/blog/swift-collections/
var names:Deque = ["Steve", "Bill", "Linus", "Bret"]
Add 'Tim' at the end of names
names.append("Tim")
Add 'Tim' at the begining of names
names.prepend("John")
Remove the first element of names
names.popFirst() // "John"
Remove the last element of names
names.popLast() // "Tim"
From page 143 of The Swift Programming Language:
You can add a new item to the end of an array by calling the array’s append method
Alternatively, add a new item to the end of an array with the addition assignment operator (+=)
Excerpt From: Apple Inc. “The Swift Programming Language.” iBooks. https://itun.es/us/jEUH0.l
To add to the solutions suggesting append, it's useful to know that this is an amortised constant time operation in many cases:
  
Complexity: Amortized O(1) unless self's storage is shared with another live array; O(count) if self does not wrap a bridged NSArray; otherwise the efficiency is unspecified.
I'm looking for a cons like operator for Swift. It should return a new immutable array with the element tacked on the end, in constant time, without changing the original array. I've not yet found a standard function that does this. I'll try to remember to report back if I find one! 
If you want to append unique object, you can expand Array struct
extension Array where Element: Equatable {
    mutating func appendUniqueObject(object: Generator.Element) {
        if contains(object) == false {
            append(object)
If the array is NSArray you can use the adding function to add any object at the end of the array, like this:
Swift 4.2
var myArray: NSArray = []
let firstElement: String = "First element"
let secondElement: String = "Second element"
// Process to add the elements to the array
myArray.adding(firstElement)
myArray.adding(secondElement)
Result:
print(myArray) 
// ["First element", "Second element"]
That is a very simple way, regards!
Example:  students = ["Ben" ,  "Ivy" ,  "Jordell"]
1) To add single elements to the end of an array, use the append(_:)
students.append(\ "Maxime" )
2) Add multiple elements at the same time by passing another array or a sequence of any kind to the append(contentsOf:) method
students.append(contentsOf: ["Shakia" , "William"])
3) To add new elements in the middle of an array by using the insert(_:at:) method for single elements
students.insert("Liam" , at:2 )
4) Using insert(contentsOf:at:) to insert multiple elements from another collection or array literal
students.insert(['Tim','TIM' at: 2 )
Swift 5.3, I believe.
The normal array wasvar myArray = ["Steve", "Bill", "Linus", "Bret"]
and you want to add "Tim" to the array, then you can use myArray.insert("Tim", at=*index*)so if you want to add it at the back of the array, then you can use myArray.append("Tim", at: 3)