Google Wave in it's current release version appears to support two types of extensions:

• Robots: Apps that run outside the wave, and are contacted by the wave (mono-directional).
• Gadgets: Apps that live inside the wave and have shared state between wave participants.

For robots the extension libraries are predictably in Python and Java and currently only use the Google App Engine as their application server.

The Gadgets on the other hand are written in HTML, CSS and Javascript, are subject to a public display condition and work primarily on the "wave" object (shared state).


[personal note]
This offers an interesting set of tools for a developer to create
extensions, which are claimed to be first class citizens within the
wave. It's reassuring that there is mention of how development has
not kept to the standard API's but is mostly compatible with them.
This may either mean a forking away from standards in the future,