How to listen to global hotkeys with Swift in an OS X app?

I don't believe you can do this in 100% Swift today. You'll need to call InstallEventHandler() or CGEventTapCreate(), and both of those require a CFunctionPointer, which can't be created in Swift. Your best plan is to use established ObjC solutions such as DDHotKey and bridge to Swift.

You can try using NSEvent.addGlobalMonitorForEventsMatchingMask(handler:), but that only makes copies of events. You can't consume them. That means the hotkey will also be passed along to the currently active app, which can cause problems. Here's an example, but I recommend the ObjC approach; it's almost certainly going to work better.

let keycode = UInt16(kVK_ANSI_X)
let keymask: NSEventModifierFlags = .CommandKeyMask | .AlternateKeyMask | .ControlKeyMask

func handler(event: NSEvent!) {
    if event.keyCode == self.keycode &&
        event.modifierFlags & self.keymask == self.keymask {
            println("PRESSED")
    }
}

// ... to set it up ...
    let options = NSDictionary(object: kCFBooleanTrue, forKey: kAXTrustedCheckOptionPrompt.takeUnretainedValue() as NSString) as CFDictionaryRef
    let trusted = AXIsProcessTrustedWithOptions(options)
    if (trusted) {
        NSEvent.addGlobalMonitorForEventsMatchingMask(.KeyDownMask, handler: self.handler)
    }

This also requires that accessibility services be approved for this app. It also doesn't capture events that are sent to your own application, so you have to either capture them with your responder chain, our use addLocalMointorForEventsMatchingMask(handler:) to add a local handler.

Since Swift 2.0, you can now pass a function pointer to C APIs.

var gMyHotKeyID = EventHotKeyID()
gMyHotKeyID.signature = OSType("swat".fourCharCodeValue)
gMyHotKeyID.id = UInt32(keyCode)

var eventType = EventTypeSpec()
eventType.eventClass = OSType(kEventClassKeyboard)
eventType.eventKind = OSType(kEventHotKeyPressed)

// Install handler.
InstallEventHandler(GetApplicationEventTarget(), {(nextHanlder, theEvent, userData) -> OSStatus in
    var hkCom = EventHotKeyID()
    GetEventParameter(theEvent, EventParamName(kEventParamDirectObject), EventParamType(typeEventHotKeyID), nil, sizeof(EventHotKeyID), nil, &hkCom)

    // Check that hkCom in indeed your hotkey ID and handle it.
}, 1, &eventType, nil, nil)

// Register hotkey.
let status = RegisterEventHotKey(UInt32(keyCode), UInt32(modifierKeys), gMyHotKeyID, GetApplicationEventTarget(), 0, &hotKeyRef)

A quick Swift 3 update for the setup:

let opts = NSDictionary(object: kCFBooleanTrue, forKey: kAXTrustedCheckOptionPrompt.takeUnretainedValue() as NSString) as CFDictionary

guard AXIsProcessTrustedWithOptions(opts) == true else { return }

NSEvent.addGlobalMonitorForEvents(matching: .keyDown, handler: self.handler)