Password Authenticated Key Exchange (PAKE) is one of those odd protocols that sounds like a great idea, but one that no one seems to be using. Even then, it seems no one can agree upon a good implementation. Secure Remote Password (SRP) is the most common implementation, found in use by Apple and 1Password; however, it is far from perfect.
I’m underqualified to explain any of those sweeping statements, so I’m going to leave it to cryptographer Matthew Green, who has two excellent articles on both PAKE and SRP. I highly recommend reading at least the first one before implementing PAKE in your application.