Use below code for an image taken from Photo Gallery and save inside photo library.

Code Support for Swift 3.1 & 4.0 version:

First, we have to do the setup for Permissions inside Project’s .plist file:-

1) Camera

<key>NSCameraUsageDescription</key>
<string>This app will use camera.</string>

2) Photo Library

<key>NSPhotoLibraryUsageDescription</key>
<string>You can select photos to attach to reports.</string>

3) Save to Photo Library

<key>NSPhotoLibraryAddUsageDescription</key>
<string>Please allow access to save photo in your photo library</string>

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We need to open .pilst file as a Source code type then add permissions inside –

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After That

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import UIKit

class ViewController: UIViewController, UINavigationControllerDelegate, UIImagePickerControllerDelegate  {

    @IBOutlet weak var imageTake: UIImageView!

  var imagePicker: UIImagePickerController!
    override func viewDidLoad() {
        super.viewDidLoad()
    }

    //MARK: - Take image
    @IBAction func takePhoto(_ sender: UIButton) {
        imagePicker =  UIImagePickerController()
        imagePicker.delegate = self
        imagePicker.sourceType = .camera
        present(imagePicker, animated: true, completion: nil)
    }

    //MARK: - Saving Image here
    @IBAction func save(_ sender: AnyObject) {
        UIImageWriteToSavedPhotosAlbum(imageTake.image!, self, #selector(image(_:didFinishSavingWithError:contextInfo:)), nil)
    }

    //MARK: - Add image to Library
    func image(_ image: UIImage, didFinishSavingWithError error: Error?, contextInfo: UnsafeRawPointer) {
        if let error = error {
            // we got back an error!
            let ac = UIAlertController(title: "Save error", message: error.localizedDescription, preferredStyle: .alert)
            ac.addAction(UIAlertAction(title: "OK", style: .default))
            present(ac, animated: true)
        } else {
            let ac = UIAlertController(title: "Saved!", message: "Your altered image has been saved to your photos.", preferredStyle: .alert)
            ac.addAction(UIAlertAction(title: "OK", style: .default))
            present(ac, animated: true)
        }
    }

    //MARK: - Done image capture here
    func imagePickerController(_ picker: UIImagePickerController, didFinishPickingMediaWithInfo info: [String : Any]) {
         imagePicker.dismiss(animated: true, completion: nil)
        imageTake.image = info[UIImagePickerControllerOriginalImage] as? UIImage
    }

}

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Swift 4.2 Code update –

 class ViewController: UIViewController, UINavigationControllerDelegate  {

    @IBOutlet weak var imageTake: UIImageView!
    var imagePicker: UIImagePickerController!

    enum ImageSource {
        case photoLibrary
        case camera
    }

    override func viewDidLoad() {
        super.viewDidLoad()
    }

    //MARK: - Take image
    @IBAction func takePhoto(_ sender: UIButton) {
        guard UIImagePickerController.isSourceTypeAvailable(.camera) else {
            selectImageFrom(.photoLibrary)
            return
        }
        selectImageFrom(.camera)
    }

    func selectImageFrom(_ source: ImageSource){
        imagePicker =  UIImagePickerController()
        imagePicker.delegate = self
        switch source {
        case .camera:
            imagePicker.sourceType = .camera
        case .photoLibrary:
            imagePicker.sourceType = .photoLibrary
        }
        present(imagePicker, animated: true, completion: nil)
    }

    //MARK: - Saving Image here
    @IBAction func save(_ sender: AnyObject) {
        guard let selectedImage = imageTake.image else {
            print("Image not found!")
            return
        }
        UIImageWriteToSavedPhotosAlbum(selectedImage, self, #selector(image(_:didFinishSavingWithError:contextInfo:)), nil)
    }

    //MARK: - Add image to Library
    @objc func image(_ image: UIImage, didFinishSavingWithError error: Error?, contextInfo: UnsafeRawPointer) {
        if let error = error {
            // we got back an error!
            showAlertWith(title: "Save error", message: error.localizedDescription)
        } else {
            showAlertWith(title: "Saved!", message: "Your image has been saved to your photos.")
        }
    }

    func showAlertWith(title: String, message: String){
        let ac = UIAlertController(title: title, message: message, preferredStyle: .alert)
        ac.addAction(UIAlertAction(title: "OK", style: .default))
        present(ac, animated: true)
    }
 }

 extension ViewController: UIImagePickerControllerDelegate{

    func imagePickerController(_ picker: UIImagePickerController, didFinishPickingMediaWithInfo info: [UIImagePickerController.InfoKey : Any]){
        imagePicker.dismiss(animated: true, completion: nil)
        guard let selectedImage = info[.originalImage] as? UIImage else {
            print("Image not found!")
            return
        }
        imageTake.image = selectedImage
    }
}