How to retrieve user's current city name?

After you setup CLLocationManager you would get location update as a latitude/longitude pair. Then you could use CLGeocoder to convert the coordinate to a user friendly place name.

Here is the sample code in Swift 4.

func locationManager(_ manager: CLLocationManager, didUpdateLocations locations: [CLLocation]) {
    if let lastLocation = locations.last {
        let geocoder = CLGeocoder()

        geocoder.reverseGeocodeLocation(lastLocation) { [weak self] (placemarks, error) in
            if error == nil {
                if let firstLocation = placemarks?[0],
                    let cityName = firstLocation.locality { // get the city name
                    self?.locationManager.stopUpdatingLocation()
                }
            }
        }
    }
}

Here is my small Swift class which helps me with getting reverse geocoded information about current location. Don't forget about NSLocationWhenInUseUsageDescription field in Info.plist.

class CurrentPlacemarkUpdater: NSObject, CLLocationManagerDelegate {

    private let locationManager = CLLocationManager()
    private let geocoder = CLGeocoder()

    private(set) var latestPlacemark: CLPlacemark?
    var onLatestPlacemarkUpdate: (() -> ())?
    var shouldStopOnUpdate: Bool = true

    func start() {
        locationManager.requestWhenInUseAuthorization()
        locationManager.desiredAccuracy = kCLLocationAccuracyKilometer
        locationManager.delegate = self
        locationManager.startUpdatingLocation()
    }

    func stop() {
        locationManager.stopUpdatingLocation()
    }

    fileprivate func updatePlacemark(for location: CLLocation) {
        geocoder.reverseGeocodeLocation(location) { [weak self] placemarks, error in
            if let placemark = placemarks?.first {
                self?.latestPlacemark = placemark
                self?.onLatestPlacemarkUpdate?()
                if self?.shouldStopOnUpdate ?? false {
                    self?.stop()
                }
            }
        }
    }

    func locationManager(_ manager: CLLocationManager, didUpdateLocations locations: [CLLocation]) {
        if let location = locations.last {
            updatePlacemark(for: location)
        }
    }

    func locationManager(_ manager: CLLocationManager, didFailWithError error: Error) {
        print("CurrentPlacemarkUpdater: \(error)")
    }

}