Example of UIPageViewController without storyboard i.e. created programmatically

AppDelegate.swift

// derived from https://www.veasoftware.com/posts/uipageviewcontroller-in-swift-xcode-62-ios-82-tutorial

import UIKit

@UIApplicationMain
class AppDelegate: UIResponder, UIApplicationDelegate {
    var window: UIWindow?
    func application(application: UIApplication, didFinishLaunchingWithOptions launchOptions: [NSObject: AnyObject]?) -> Bool {
        window = UIWindow(frame: UIScreen.mainScreen().bounds)
        window!.rootViewController = ViewController()
        window!.makeKeyAndVisible()
        return true
    }
}


struct Page {
    var title: String
    var image: String
}


let PAGES = [
    Page(title: "Explore", image: "https://i.imgur.com/FGknouw.png"),
    Page(title: "Today Widget", image: "https://i.imgur.com/feuwhEc.png"),
]


class ViewController: UIViewController {
    var pageViewController: UIPageViewController!

    override func viewDidLoad() {
        super.viewDidLoad()
        self.view.backgroundColor = UIColor.whiteColor()

        let pc = UIPageControl.appearance()
        pc.pageIndicatorTintColor = UIColor.lightGrayColor()
        pc.currentPageIndicatorTintColor = UIColor.blackColor()
        pc.backgroundColor = UIColor.whiteColor()

        let btn = UIButton(type: .System)
        btn.setTitle("Restart", forState: .Normal)
        btn.addTarget(self, action: "restartAction:", forControlEvents: .TouchUpInside)

        self.pageViewController = UIPageViewController(transitionStyle: .Scroll, navigationOrientation: .Horizontal, options: nil)
        self.pageViewController.dataSource = self
        self.restartAction(self)
        self.addChildViewController(self.pageViewController)

        let views = [
            "pg": self.pageViewController.view,
            "btn": btn,
        ]
        for (_, v) in views {
            v.translatesAutoresizingMaskIntoConstraints = false
            self.view.addSubview(v)
        }
        NSLayoutConstraint.activateConstraints(
            [NSLayoutConstraint(
                item: btn,
                attribute: .CenterX,
                relatedBy: .Equal,
                toItem: self.view,
                attribute: .CenterX,
                multiplier: 1,
                constant: 0),
            ] +
            NSLayoutConstraint.constraintsWithVisualFormat("H:|[pg]|",               options: .AlignAllCenterX, metrics: [:], views: views) +
            NSLayoutConstraint.constraintsWithVisualFormat("V:|-30-[pg]-[btn]-15-|", options: .AlignAllCenterX, metrics: [:], views: views)
        )
        self.pageViewController.didMoveToParentViewController(self)
    }

    func restartAction(sender: AnyObject) {
        self.pageViewController.setViewControllers([self.viewControllerAtIndex(0)], direction: .Forward, animated: true, completion: nil)
    }

    func viewControllerAtIndex(index: Int) -> ContentViewController {
        if (PAGES.count == 0) || (index >= PAGES.count) {
            return ContentViewController()
        }
        let vc = ContentViewController()
        vc.pageIndex = index
        return vc
    }

}

// MARK: - Page View Controller Data Source

extension ViewController: UIPageViewControllerDataSource {
    func pageViewController(pageViewController: UIPageViewController, viewControllerBeforeViewController viewController: UIViewController) -> UIViewController? {
        let vc = viewController as! ContentViewController
        var index = vc.pageIndex as Int
        if (index == 0 || index == NSNotFound) {
            return nil
        }
        index--
        return self.viewControllerAtIndex(index)
    }

    func pageViewController(pageViewController: UIPageViewController, viewControllerAfterViewController viewController: UIViewController) -> UIViewController? {
        let vc = viewController as! ContentViewController
        var index = vc.pageIndex as Int
        if (index == NSNotFound) {
            return nil
        }
        index++
        if (index == PAGES.count) {
            return nil
        }
        return self.viewControllerAtIndex(index)
    }

    func presentationCountForPageViewController(pageViewController: UIPageViewController) -> Int {
        return PAGES.count
    }

    func presentationIndexForPageViewController(pageViewController: UIPageViewController) -> Int {
        return 0
    }
}


class ContentViewController: UIViewController {
    var pageIndex: Int!

    override func viewDidLoad() {
        super.viewDidLoad()

        let lb = UILabel()
        lb.textAlignment = .Center
        lb.text = PAGES[self.pageIndex].title

        let iv = UIImageView()
        iv.contentMode = .ScaleAspectFit
        ImageLoader.sharedLoader.imageForUrl(PAGES[self.pageIndex].image) { (image, url) -> () in
            iv.image = image
        }

        let views = [
            "iv": iv,
            "lb": lb,
        ]
        for (_, v) in views {
            v.translatesAutoresizingMaskIntoConstraints = false
            self.view.addSubview(v)
        }
        NSLayoutConstraint.activateConstraints(
            NSLayoutConstraint.constraintsWithVisualFormat("H:|-[lb]-|",      options: .AlignAllCenterX, metrics: [:], views: views) +
            NSLayoutConstraint.constraintsWithVisualFormat("H:|-[iv]-|",      options: .AlignAllCenterX, metrics: [:], views: views) +
            NSLayoutConstraint.constraintsWithVisualFormat("V:|-[lb]-[iv]-|", options: .AlignAllCenterX, metrics: [:], views: views)
        )
    }
}


class ImageLoader {
    // source and license: https://github.com/natelyman/SwiftImageLoader

    let cache = NSCache()

    class var sharedLoader : ImageLoader {
        struct Static {
            static let instance : ImageLoader = ImageLoader()
        }
        return Static.instance
    }

    func imageForUrl(urlString: String, completionHandler: (image: UIImage?, url: String) -> ()) {
        dispatch_async(dispatch_get_global_queue(DISPATCH_QUEUE_PRIORITY_BACKGROUND, 0), {()in
            let data = self.cache.objectForKey(urlString) as? NSData

            if let goodData = data {
                let image = UIImage(data: goodData)
                dispatch_async(dispatch_get_main_queue(), {() in
                    completionHandler(image: image, url: urlString)
                })
                return
            }

            let downloadTask: NSURLSessionDataTask = NSURLSession.sharedSession().dataTaskWithURL(NSURL(string: urlString)!, completionHandler: {(data: NSData?, response: NSURLResponse?, error: NSError?) -> Void in
                if (error != nil) {
                    completionHandler(image: nil, url: urlString)
                    return
                }

                if let data = data {
                    let image = UIImage(data: data)
                    self.cache.setObject(data, forKey: urlString)
                    dispatch_async(dispatch_get_main_queue(), {() in
                        completionHandler(image: image, url: urlString)
                    })
                    return
                }
                
            })
            downloadTask.resume()
        })
    }
}

Info.plist needs presence of Launch screen interface file base name (which can be just an empty string) in order to support all screen sizes.

@adreif where is index declaration?

The second problem which allows to use this code as the example only and requires significant changes is:
extension ViewController: UIPageViewControllerDataSource {
You write an extension over all view controllers even those ones where you have never needed page view controller

@gerchicov-bp Actually, he's extending the custom class ViewController, not UIViewController