UIBarButtonItem in navigation bar programmatically?

Custom button image without setting button frame:

You can use init(image: UIImage?, style: UIBarButtonItemStyle, target: Any?, action: Selector?) to initializes a new item using the specified image and other properties.

let button1 = UIBarButtonItem(image: UIImage(named: "imagename"), style: .plain, target: self, action: Selector("action")) // action:#selector(Class.MethodName) for swift 3
self.navigationItem.rightBarButtonItem  = button1

Check this Apple Doc. reference

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UIBarButtonItem with custom button image using button frame

FOR Swift 3.0

    let btn1 = UIButton(type: .custom)
    btn1.setImage(UIImage(named: "imagename"), for: .normal)
    btn1.frame = CGRect(x: 0, y: 0, width: 30, height: 30)
    btn1.addTarget(self, action: #selector(Class.Methodname), for: .touchUpInside)
    let item1 = UIBarButtonItem(customView: btn1)

    let btn2 = UIButton(type: .custom)
    btn2.setImage(UIImage(named: "imagename"), for: .normal)
    btn2.frame = CGRect(x: 0, y: 0, width: 30, height: 30)
    btn2.addTarget(self, action: #selector(Class.MethodName), for: .touchUpInside)
    let item2 = UIBarButtonItem(customView: btn2)  

    self.navigationItem.setRightBarButtonItems([item1,item2], animated: true)

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FOR Swift 2.0 and older

let btnName = UIButton()
btnName.setImage(UIImage(named: "imagename"), forState: .Normal)
btnName.frame = CGRectMake(0, 0, 30, 30)
btnName.addTarget(self, action: Selector("action"), forControlEvents: .TouchUpInside)

//.... Set Right/Left Bar Button item
let rightBarButton = UIBarButtonItem()
rightBarButton.customView = btnName
self.navigationItem.rightBarButtonItem = rightBarButton

Or simply use init(customView:) like

 let rightBarButton = UIBarButtonItem(customView: btnName)
 self.navigationItem.rightBarButtonItem = rightBarButton

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For System UIBarButtonItem

let camera = UIBarButtonItem(barButtonSystemItem: .Camera, target: self, action: Selector("btnOpenCamera"))
self.navigationItem.rightBarButtonItem = camera

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For set more then 1 items use rightBarButtonItems or for left side leftBarButtonItems

let btn1 = UIButton()
btn1.setImage(UIImage(named: "img1"), forState: .Normal)
btn1.frame = CGRectMake(0, 0, 30, 30)
btn1.addTarget(self, action: Selector("action1:"), forControlEvents: .TouchUpInside)
let item1 = UIBarButtonItem()
item1.customView = btn1

let btn2 = UIButton()
btn2.setImage(UIImage(named: "img2"), forState: .Normal)
btn2.frame = CGRectMake(0, 0, 30, 30)
btn2.addTarget(self, action: Selector("action2:"), forControlEvents: .TouchUpInside)
let item2 = UIBarButtonItem()
item2.customView = btn2

self.navigationItem.rightBarButtonItems = [item1,item2]

Using setLeftBarButtonItem or setRightBarButtonItem

let btn1 = UIButton()
btn1.setImage(UIImage(named: "img1"), forState: .Normal)
btn1.frame = CGRectMake(0, 0, 30, 30)
btn1.addTarget(self, action: Selector("action1:"), forControlEvents: .TouchUpInside)
self.navigationItem.setLeftBarButtonItem(UIBarButtonItem(customView: btn1), animated: true);

For swift >= 2.2 action should be #selector(Class.MethodName) ... for e.g. btnName.addTarget(self, action: #selector(Class.MethodName), forControlEvents: .TouchUpInside)

It's much easier with Swift 4 or Swift 4.2

inside your ViewDidLoad method, define your button and add it to the navigation bar.

override func viewDidLoad() {
    super.viewDidLoad()

    let logoutBarButtonItem = UIBarButtonItem(title: "Logout", style: .done, target: self, action: #selector(logoutUser))
    self.navigationItem.rightBarButtonItem  = logoutBarButtonItem

}

then you need to define the function that you mentioned inside action parameter as below

@objc func logoutUser(){
     print("clicked")
}

You need to add the @objc prefix as it's still making use of the legacy stuff (Objective C).