Check the AVAuthorizationStatus and handle the cases properly.

NSString *mediaType = AVMediaTypeVideo;
AVAuthorizationStatus authStatus = [AVCaptureDevice authorizationStatusForMediaType:mediaType];
if(authStatus == AVAuthorizationStatusAuthorized) {
  // do your logic
} else if(authStatus == AVAuthorizationStatusDenied){
  // denied
} else if(authStatus == AVAuthorizationStatusRestricted){
  // restricted, normally won't happen
} else if(authStatus == AVAuthorizationStatusNotDetermined){
  // not determined?!
  [AVCaptureDevice requestAccessForMediaType:mediaType completionHandler:^(BOOL granted) {
    if(granted){
      NSLog(@"Granted access to %@", mediaType);
    } else {
      NSLog(@"Not granted access to %@", mediaType);
    }
  }];
} else {
  // impossible, unknown authorization status
}

Swift 4 and newer

Make sure to:

import AVFoundation

The code below checks for all possible permission states:

let cameraMediaType = AVMediaType.video
let cameraAuthorizationStatus = AVCaptureDevice.authorizationStatus(for: cameraMediaType)
    
switch cameraAuthorizationStatus {
case .denied: break
case .authorized: break
case .restricted: break

case .notDetermined:
    // Prompting user for the permission to use the camera.
    AVCaptureDevice.requestAccess(for: cameraMediaType) { granted in
        if granted {
            print("Granted access to \(cameraMediaType)")
        } else {
            print("Denied access to \(cameraMediaType)")
        }
    }
}

Since iOS 10 you need to specify NSCameraUsageDescription key in your Info.plist to be able ask for camera access, otherwise your app will crash at runtime. See APIs Requiring Usage Descriptions.


An interesting sidenote from Apple Developer forum:

The system actually kills your app if the user toggles your app's access to camera in Settings. The same applies to any protected dataclass in the Settings→Privacy section.


Swift Solution

extension AVCaptureDevice {
    enum AuthorizationStatus {
        case justDenied
        case alreadyDenied
        case restricted
        case justAuthorized
        case alreadyAuthorized
        case unknown
    }

    class func authorizeVideo(completion: ((AuthorizationStatus) -> Void)?) {
        AVCaptureDevice.authorize(mediaType: AVMediaType.video, completion: completion)
    }

    class func authorizeAudio(completion: ((AuthorizationStatus) -> Void)?) {
        AVCaptureDevice.authorize(mediaType: AVMediaType.audio, completion: completion)
    }

    private class func authorize(mediaType: AVMediaType, completion: ((AuthorizationStatus) -> Void)?) {
        let status = AVCaptureDevice.authorizationStatus(for: mediaType)
        switch status {
        case .authorized:
            completion?(.alreadyAuthorized)
        case .denied:
            completion?(.alreadyDenied)
        case .restricted:
            completion?(.restricted)
        case .notDetermined:
            AVCaptureDevice.requestAccess(for: mediaType, completionHandler: { (granted) in
                DispatchQueue.main.async {
                    if granted {
                        completion?(.justAuthorized)
                    } else {
                        completion?(.justDenied)
                    }
                }
            })
        @unknown default:
            completion?(.unknown)
        }
    }
}

And then in order to use it you do

AVCaptureDevice.authorizeVideo(completion: { (status) in
   //Your work here
})

As an addition to the answer from @Raptor the following should be mentioned. You may receive the following error starting with iOS 10: This application is modifying the autolayout engine from a background thread after the engine was accessed from the main thread. This can lead to engine corruption and weird crashes.

To fix this, make sure you handle the results from the main thread as follows (Swift 3):

private func showCameraPermissionPopup() {
    let cameraMediaType = AVMediaTypeVideo
    let cameraAuthorizationStatus = AVCaptureDevice.authorizationStatus(forMediaType: cameraMediaType)

    switch cameraAuthorizationStatus {
    case .denied:
        NSLog("cameraAuthorizationStatus=denied")
        break
    case .authorized:
        NSLog("cameraAuthorizationStatus=authorized")
        break
    case .restricted:
        NSLog("cameraAuthorizationStatus=restricted")
        break
    case .notDetermined:
        NSLog("cameraAuthorizationStatus=notDetermined")

        // Prompting user for the permission to use the camera.
        AVCaptureDevice.requestAccess(forMediaType: cameraMediaType) { granted in
            DispatchQueue.main.sync {
                if granted {
                    // do something
                } else {
                    // do something else
                }
            }
        }
    }
}