If you build with the latest beta iOS 13.4 and Xcode 11.4, the accepted answer won't work anymore. I've found another way, maybe it's just a bug in the beta software, but I'm writing it down there, just in case

(swift 5)

import UIKit

class TransparentNavBar :UINavigationBar {
    override func awakeFromNib() {
        super.awakeFromNib()
        self.setBackgroundImage(UIImage(), for: .default)
        self.shadowImage = UIImage()
        self.isTranslucent = true
        self.backgroundColor = .clear
        if #available(iOS 13.0, *) {
            self.standardAppearance.backgroundColor = .clear
            self.standardAppearance.backgroundEffect = .none
            self.standardAppearance.shadowColor = .clear
        }
    }
}

This seems to work:

@implementation UINavigationBar (custom)
- (void)drawRect:(CGRect)rect {}
@end

navigationController.navigationBar.backgroundColor = [UIColor clearColor];

Solution - Swift 5 - iOS 13+

According to the documentation, in your UIViewController subclass:

override func viewDidLoad()
{
    super.viewDidLoad()
    
    let appearance = UINavigationBarAppearance()
    appearance.configureWithTransparentBackground()
    //appearance.backgroundColor = UIColor.clear
    
    navigationItem.compactAppearance = appearance
    navigationItem.scrollEdgeAppearance = appearance
    navigationItem.standardAppearance = appearance
    
    //...
}

Just to be clear, this makes the UINavigationBar completely transparent. The bar button items are still visible and work properly.

What didn't work

override func viewDidLoad()
{
    super.viewDidLoad()
    
    navigationController?.navigationBar.isTranslucent = true
    navigationController?.navigationBar.isOpaque = false

    //...
}

This made me realize I didn't actually know the difference between transparent and translucent RIP.

References

https://developer.apple.com/documentation/uikit/uinavigationcontroller/customizing_your_app_s_navigation_bar

https://www.lexico.com/en/definition/transparent

https://www.lexico.com/en/definition/translucent

Update 08/10/2021

Changing the navigationItem bar buttons after setting the appearance in the way I provided will reset the appearance and you'll have to do it again.

for Swift 3.0:

override func viewDidLoad() {
    super.viewDidLoad()

    navigationController?.navigationBar.setBackgroundImage(UIImage(), for: .default)
    navigationController?.navigationBar.shadowImage = UIImage()
    navigationController?.navigationBar.isTranslucent = true
}

Try the following piece of code:

self.navigationController.navigationBar.translucent = YES;

In Swift 4.2

self.navigationController?.navigationBar.setBackgroundImage(UIImage(), for: .default)
self.navigationController?.navigationBar.shadowImage = UIImage()
self.navigationController?.navigationBar.isTranslucent = true

(in viewWillAppear), and then in viewWillDisappear, to undo it, put

self.navigationController?.navigationBar.shadowImage = nil
self.navigationController?.navigationBar.isTranslucent = false

This worked with Swift 5.

// Clear the background image.
navigationController?.navigationBar.setBackgroundImage(UIImage(), for: .default)

// Clear the shadow image.
navigationController?.navigationBar.shadowImage = UIImage()

// Ensure the navigation bar is translucent.
navigationController?.navigationBar.isTranslucent = true

Do you mean entirely transparent, or using the translucent-black style seen in the Photos app? The latter you can accomplish by setting its barStyle property to UIBarStyleBlackTranslucent. The former... I'm not sure about. If you want the items on it to still be visible, you might have to do some digging around in the bar's view hierarchy and remove the view containing its background.

Check RRViewControllerExtension, which is dedicated on UINavigation bar appearance management.

with RRViewControllerExtension in your project, you just need to override

-(BOOL)prefersNavigationBarTransparent;

in you viewcontroller.

extension UINavigationBar {
var isTransperent: Bool {
        get {
            return false // Just to satisfy property
        }
        set {
            if newValue {
                self.shadowImage = UIImage()
                self.isTranslucent = true
                self.setBackgroundImage(UIImage(), for: .default)
            } else {
                self.shadowImage = UIImage()
                self.isTranslucent = false
                self.setBackgroundImage(nil, for: .default)
            }
        }
    }
}