Or does anyone know of any mobile applications that have done it well? Thanks!
One of my startups leveraged mobile videoconferencing to provide real time coaching lessons. We implemented https://tokbox.com. It is platform independent, relatively easy to implement, and highly scalable.
They have plenty of customer success stories to back it up as well. https://tokbox.com/customer-stories
If you have additional questions around implementation, I am here to chat.
Best of luck!
I assume you mean a mobile web application (in-browser based), as opposed to a native apps? If that is the case you have to go no further then searching for deeply functional mobile and responsive sites like mobile.twitter.com .
As far as OS-agnostic mobile web applications for video conferencing—no better place to start then Skype.