Swift 2 example for user4806509 from http://stackoverflow.com/a/28755141/1271826

AsyncCompletionHandlers.Swift

// Swift 2

override func viewDidLoad() {
    super.viewDidLoad()
    
    perform1 {
        self.perform2 {
            self.perform3 {
                print("everything done")
            }
        }
    }
}

func perform1(completionHandler: () -> Void) {
    doSomethingAsynchronously() {
        completionHandler()
    }
}

func perform2(completionHandler: () -> Void) {
    doSomethingElseAsynchronously() {
        completionHandler()
    }
}

func perform3(completionHandler: () -> Void) {
    doSomethingCompletelyDifferentAsynchronously() {
        completionHandler()
    }
}

// clearly, you would do something more interesting than simply dispatching something after 4 seconds

func doSomethingAsynchronously(completionHandler: () -> Void) {
    print("starting doSomethingAsynchronously")
    dispatch_after(dispatch_time(DISPATCH_TIME_NOW, Int64(4 * NSEC_PER_SEC)), dispatch_get_main_queue()) {
        completionHandler()
        print("finished doSomethingAsynchronously")
    }
}

func doSomethingElseAsynchronously(completionHandler: () -> Void) {
    print("starting doSomethingElseAsynchronously")
    dispatch_after(dispatch_time(DISPATCH_TIME_NOW, Int64(4 * NSEC_PER_SEC)), dispatch_get_main_queue()) {
        completionHandler()
        print("finished doSomethingElseAsynchronously")
    }
}

func doSomethingCompletelyDifferentAsynchronously(completionHandler: () -> Void) {
    print("starting doSomethingCompletelyDifferentAsynchronously")
    dispatch_after(dispatch_time(DISPATCH_TIME_NOW, Int64(4 * NSEC_PER_SEC)), dispatch_get_main_queue()) {
        completionHandler()
        print("finished doSomethingCompletelyDifferentAsynchronously")
    }
}

When I originally wrote this example, I assumed that, in perform1, perform2, and perform3, you wouldn't call your own doSomething... asynchronous methods, but rather call some some existing asynchronous API (e.g. geocode requests which shouldn't be run concurrently, network requests where each request is contingent upon the value retrieved by the prior request, etc.).

This example, where we have trivial methods calling other trivial methods is somewhat contrived. Also, you'd generally be passing parameters into these various methods, as well as having the closures passing data back. But hopefully this illustrates the basic idea.