Users are reporting that Microsoft's Skype Preview app for Windows 10 is crashing, on both PC and Mobile. The issue appears to be with version 18.104.22.168, which is only available to Windows Insiders in the Release Preview, Slow, and Fast rings. Production ring (non-Insiders) users are still on 22.214.171.124, so they shouldn't have a problem.
Indeed, we experienced the exact same issue with our test units. It's currently unknown just how widespread the issue is, but it seems to be just about everyone. In fact, you can find plenty of people complaining about it on AppRaisin and in the Skype forums.
The obvious solution on a PC is to simply download the desktop app. Since you're on the Anniversary Update, the desktop and UWP apps tend to act a bit wonky with each other, and your PC might try to force you out of Skype and into Skype Preview, which would also close. Your best bet might be to simply uninstall Skype Preview.
For Windows 10 Mobile users, the solution isn't so simple. There is no alternative app for Skype, and it's not so easy to go back to an earlier version. If you're on the Release Preview ring, you can easily switch to Production, uninstall the app, and reinstall it. For Slow and Fast users, you might want to just wait for the next update, which shouldn't be far away.
Update: Some users are also reporting problems on the Production ring in response to this article. We've reached out to Microsoft for a statement.
Update: With some help from @TheCheekyTaurus, it appears that this update was available in the Production ring and has since been pulled.
Update: Microsoft has confirmed that this issue only affected Insiders, and has issued a fix.