Make a UIWebView as height as its content with Auto Layout

To make this work you have to do the following steps:

1. Connect the UIWebView from the nib to an outlet in your view controller
2. Disable scrolling in the web view
3. Set the constraints on the UIScrollView, the UIView on top of the web view (In my example I omitted all the labels in that view) and the UIWebView.
4. Connect the UIWebView's height constraint to an outlet in your view controller.
5. Set the view controller as UIWebViewDelegate
6. In webViewDidFinishLoad set the height constraint's constant to the height of the contentSize of the scroll view inside the web view.
7. Start Key-Value Observing on the contentSize to change the height, when height of the web view has to change because segments of the webpage change their size without reloading the page (like accordeons, or menus).

I won't explain the constraints in detail as you seem to already have figured them out yourself. Here is a screenshot of the constraints:

So, here is the code:

import UIKit

var MyObservationContext = 0

class ViewController: UIViewController {

    @IBOutlet weak var webview: UIWebView!
    @IBOutlet weak var webviewHeightConstraint: NSLayoutConstraint!
    var observing = false

    override func viewDidLoad() {
        super.viewDidLoad()
        webview.scrollView.scrollEnabled = false
        webview.delegate = self
        webview.loadRequest(NSURLRequest(URL: NSURL(string: "https://www.google.de/intl/de/policies/terms/regional.html")!))
    }

    deinit {
        stopObservingHeight()
    }

    func startObservingHeight() {
        let options = NSKeyValueObservingOptions([.New])
        webview.scrollView.addObserver(self, forKeyPath: "contentSize", options: options, context: &MyObservationContext)
        observing = true;
    }

    func stopObservingHeight() {
        webview.scrollView.removeObserver(self, forKeyPath: "contentSize", context: &MyObservationContext)
        observing = false
    }

    override func observeValueForKeyPath(keyPath: String?, ofObject object: AnyObject?, change: [String : AnyObject]?, context: UnsafeMutablePointer<Void>) {
        guard let keyPath = keyPath else {
            super.observeValueForKeyPath(nil, ofObject: object, change: change, context: context)
            return
        }
        switch (keyPath, context) {
        case("contentSize", &MyObservationContext):
            webviewHeightConstraint.constant = webview.scrollView.contentSize.height
        default:
            super.observeValueForKeyPath(keyPath, ofObject: object, change: change, context: context)
        }
    }
}

extension ViewController: UIWebViewDelegate {
    func webViewDidFinishLoad(webView: UIWebView) {
        print(webView.request?.URL)
        webviewHeightConstraint.constant = webview.scrollView.contentSize.height
        if (!observing) {
            startObservingHeight()
        }
    }
}

Great solution from @joern. Works fine for me. I just adapted his code to swift 3.

Swift 3 Update Code :

I also modify stopObservingHeight to prevent remove observer before it was create.

import UIKit

var MyObservationContext = 0

class ViewController: UIViewController {

    @IBOutlet weak var webview: UIWebView!
    @IBOutlet weak var webviewHeightConstraint: NSLayoutConstraint!
    var observing = false

    override func viewDidLoad() {
        super.viewDidLoad()
        webview.scrollView.isScrollEnabled = false
        webview.delegate = self
        webview.loadRequest(NSURLRequest(URL: NSURL(string: "https://www.google.de/intl/de/policies/terms/regional.html")!))
    }

    deinit {
        stopObservingHeight()
    }

    func startObservingHeight() {
        let options = NSKeyValueObservingOptions([.new])
        webview.scrollView.addObserver(self, forKeyPath: "contentSize", options: options, context: &MyObservationContext)
        observing = true;
    }

    func stopObservingHeight() {
        if observing {
            webView.scrollView.removeObserver(self, forKeyPath: "contentSize", context: &MyObservationContext)            
            observing = false
        }
    }

    override func observeValue(forKeyPath keyPath: String?, of object: Any?, change: [NSKeyValueChangeKey : Any]?, context: UnsafeMutableRawPointer?) {
        guard let keyPath = keyPath,
            let context = context else {
            super.observeValue(forKeyPath: nil, of: object, change: change, context: nil)
            return
        }
        switch (keyPath, context) {
        case("contentSize", &MyObservationContext):
            webviewHeightConstraint.constant = webview.scrollView.contentSize.height
        default:
            super.observeValue(forKeyPath: keyPath, of: object, change: change, context: context)
        }
    }
}

extension ViewController: UIWebViewDelegate {
    func webViewDidFinishLoad(_ webView: UIWebView) {
        print(webView.request?.url ?? "nil")
        webviewHeightConstraint.constant = webview.scrollView.contentSize.height
        if (!observing) {
            startObservingHeight()
        }
    }
}