Updated for Swift 4.2

let appVersion = Bundle.main.infoDictionary?["CFBundleShortVersionString"] as? String


As pointed out by @azdev on the new version of Xcode you will get a compile error for trying my previous solution, to solve this just edit it as suggested to unwrap the bundle dictionary using a !

let nsObject: AnyObject? = Bundle.main.infoDictionary!["CFBundleShortVersionString"]

End Edit

Just use the same logic than in Objective-C but with some small changes

//First get the nsObject by defining as an optional anyObject
let nsObject: AnyObject? = NSBundle.mainBundle().infoDictionary["CFBundleShortVersionString"]

//Then just cast the object as a String, but be careful, you may want to double check for nil
let version = nsObject as! String