How would I create a UIAlertView in Swift?

From the UIAlertView class:

// UIAlertView is deprecated. Use UIAlertController with a
preferredStyle of UIAlertControllerStyleAlert instead

On iOS 8, you can do this:

let alert = UIAlertController(title: "Alert", message: "Message", preferredStyle: UIAlertControllerStyle.Alert)
alert.addAction(UIAlertAction(title: "Click", style: UIAlertActionStyle.Default, handler: nil))
self.presentViewController(alert, animated: true, completion: nil)

Now UIAlertController is a single class for creating and interacting with what we knew as UIAlertViews and UIActionSheets on iOS 8.

Edit: To handle actions:

alert.addAction(UIAlertAction(title: "OK", style: .Default, handler: { action in
    switch action.style{
    case .Default:
        print("default")
        
    case .Cancel:
        print("cancel")
        
    case .Destructive:
        print("destructive")
    }
}}))

Edit for Swift 3:

let alert = UIAlertController(title: "Alert", message: "Message", preferredStyle: UIAlertControllerStyle.alert)
alert.addAction(UIAlertAction(title: "Click", style: UIAlertActionStyle.default, handler: nil))
self.present(alert, animated: true, completion: nil)

Edit for Swift 4.x:

let alert = UIAlertController(title: "Alert", message: "Message", preferredStyle: .alert)
alert.addAction(UIAlertAction(title: "OK", style: .default, handler: { action in
    switch action.style{
        case .default:
        print("default")
        
        case .cancel:
        print("cancel")
        
        case .destructive:
        print("destructive")
        
    }
}))
self.present(alert, animated: true, completion: nil)