How to have stored properties in Swift, the same way I had on Objective-C?

As in Objective-C, you can't add stored property to existing classes. If you're extending an Objective-C class (UIView is definitely one), you can still use Associated Objects to emulate stored properties:

for Swift 1

import ObjectiveC

private var xoAssociationKey: UInt8 = 0

extension UIView {
    var xo: PFObject! {
        get {
            return objc_getAssociatedObject(self, &xoAssociationKey) as? PFObject
        }
        set(newValue) {
            objc_setAssociatedObject(self, &xoAssociationKey, newValue, objc_AssociationPolicy(OBJC_ASSOCIATION_RETAIN))
        }
    }
}

The association key is a pointer that should be the unique for each association. For that, we create a private global variable and use it's memory address as the key with the & operator. See the Using Swift with Cocoa and Objective-C on more details how pointers are handled in Swift.

UPDATED for Swift 2 and 3

import ObjectiveC

private var xoAssociationKey: UInt8 = 0

extension UIView {
    var xo: PFObject! {
        get {
            return objc_getAssociatedObject(self, &xoAssociationKey) as? PFObject
        }
        set(newValue) {
            objc_setAssociatedObject(self, &xoAssociationKey, newValue, objc_AssociationPolicy.OBJC_ASSOCIATION_RETAIN)
        }
    }
}

UPDATED for Swift 4

In Swift 4, it's much more simple. The Holder struct will contain the private value that our computed property will expose to the world, giving the illusion of a stored property behaviour instead.

Source

extension UIViewController {
    struct Holder {
        static var _myComputedProperty:Bool = false
    }
    var myComputedProperty:Bool {
        get {
            return Holder._myComputedProperty
        }
        set(newValue) {
            Holder._myComputedProperty = newValue
        }
    }
}

So I think I found a method that works cleaner than the ones above because it doesn't require any global variables. I got it from here: http://nshipster.com/swift-objc-runtime/

The gist is that you use a struct like so:

extension UIViewController {
    private struct AssociatedKeys {
        static var DescriptiveName = "nsh_DescriptiveName"
    }

    var descriptiveName: String? {
        get {
            return objc_getAssociatedObject(self, &AssociatedKeys.DescriptiveName) as? String
        }
        set {
            if let newValue = newValue {
                objc_setAssociatedObject(
                    self,
                    &AssociatedKeys.DescriptiveName,
                    newValue as NSString?,
                    UInt(OBJC_ASSOCIATION_RETAIN_NONATOMIC)
                )
            }
        }
    }
}

UPDATE for Swift 2

private struct AssociatedKeys {
    static var displayed = "displayed"
}

//this lets us check to see if the item is supposed to be displayed or not
var displayed : Bool {
    get {
        guard let number = objc_getAssociatedObject(self, &AssociatedKeys.displayed) as? NSNumber else {
            return true
        }
        return number.boolValue
    }

    set(value) {
        objc_setAssociatedObject(self,&AssociatedKeys.displayed,NSNumber(bool: value),objc_AssociationPolicy.OBJC_ASSOCIATION_RETAIN_NONATOMIC)
    }
}

Associated objects API is a bit cumbersome to use. You can remove most of the boilerplate with a helper class.

public final class ObjectAssociation<T: AnyObject> {

    private let policy: objc_AssociationPolicy

    /// - Parameter policy: An association policy that will be used when linking objects.
    public init(policy: objc_AssociationPolicy = .OBJC_ASSOCIATION_RETAIN_NONATOMIC) {

        self.policy = policy
    }

    /// Accesses associated object.
    /// - Parameter index: An object whose associated object is to be accessed.
    public subscript(index: AnyObject) -> T? {

        get { return objc_getAssociatedObject(index, Unmanaged.passUnretained(self).toOpaque()) as! T? }
        set { objc_setAssociatedObject(index, Unmanaged.passUnretained(self).toOpaque(), newValue, policy) }
    }
}

Provided that you can "add" a property to objective-c class in a more readable manner:

extension SomeType {

    private static let association = ObjectAssociation<NSObject>()

    var simulatedProperty: NSObject? {

        get { return SomeType.association[self] }
        set { SomeType.association[self] = newValue }
    }
}

As for the solution:

extension CALayer {

    private static let initialPathAssociation = ObjectAssociation<CGPath>()
    private static let shapeLayerAssociation = ObjectAssociation<CAShapeLayer>()

    var initialPath: CGPath! {
        get { return CALayer.initialPathAssociation[self] }
        set { CALayer.initialPathAssociation[self] = newValue }
    }

    var shapeLayer: CAShapeLayer? {
        get { return CALayer.shapeLayerAssociation[self] }
        set { CALayer.shapeLayerAssociation[self] = newValue }
    }
}

The solution pointed out by jou doesn't support value types, this works fine with them as well

Wrappers

import ObjectiveC

final class Lifted<T> {
    let value: T
    init(_ x: T) {
        value = x
    }
}

private func lift<T>(x: T) -> Lifted<T>  {
    return Lifted(x)
}

func setAssociatedObject<T>(object: AnyObject, value: T, associativeKey: UnsafePointer<Void>, policy: objc_AssociationPolicy) {
    if let v: AnyObject = value as? AnyObject {
        objc_setAssociatedObject(object, associativeKey, v,  policy)
    }
    else {
        objc_setAssociatedObject(object, associativeKey, lift(value),  policy)
    }
}

func getAssociatedObject<T>(object: AnyObject, associativeKey: UnsafePointer<Void>) -> T? {
    if let v = objc_getAssociatedObject(object, associativeKey) as? T {
        return v
    }
    else if let v = objc_getAssociatedObject(object, associativeKey) as? Lifted<T> {
        return v.value
    }
    else {
        return nil
    }
}

A possible Class extension (Example of usage):

extension UIView {

    private struct AssociatedKey {
        static var viewExtension = "viewExtension"
    }

    var referenceTransform: CGAffineTransform? {
        get {
            return getAssociatedObject(self, associativeKey: &AssociatedKey.viewExtension)
        }

        set {
            if let value = newValue {
                setAssociatedObject(self, value: value, associativeKey: &AssociatedKey.viewExtension, policy: objc_AssociationPolicy.OBJC_ASSOCIATION_RETAIN_NONATOMIC)
            }
        }
    }
}

This is really such a great solution, I wanted to add another usage example that included structs and values that are not optionals. Also, the AssociatedKey values can be simplified.

struct Crate {
    var name: String
}

class Box {
    var name: String

    init(name: String) {
        self.name = name
    }
}

extension UIViewController {

    private struct AssociatedKey {
        static var displayed:   UInt8 = 0
        static var box:         UInt8 = 0
        static var crate:       UInt8 = 0
    }

    var displayed: Bool? {
        get {
            return getAssociatedObject(self, associativeKey: &AssociatedKey.displayed)
        }

        set {
            if let value = newValue {
                setAssociatedObject(self, value: value, associativeKey: &AssociatedKey.displayed, policy: objc_AssociationPolicy.OBJC_ASSOCIATION_RETAIN_NONATOMIC)
            }
        }
    }

    var box: Box {
        get {
            if let result:Box = getAssociatedObject(self, associativeKey: &AssociatedKey.box) {
                return result
            } else {
                let result = Box(name: "")
                self.box = result
                return result
            }
        }

        set {
            setAssociatedObject(self, value: newValue, associativeKey: &AssociatedKey.box, policy: objc_AssociationPolicy.OBJC_ASSOCIATION_RETAIN_NONATOMIC)
        }
    }

    var crate: Crate {
        get {
            if let result:Crate = getAssociatedObject(self, associativeKey: &AssociatedKey.crate) {
                return result
            } else {
                let result = Crate(name: "")
                self.crate = result
                return result
            }
        }

        set {
            setAssociatedObject(self, value: newValue, associativeKey: &AssociatedKey.crate, policy: objc_AssociationPolicy.OBJC_ASSOCIATION_RETAIN_NONATOMIC)
        }
    }
}