Javascript Git client
This https://github.com/creationix/js-git is and will be the future!
It is backed by a kickstarter campaign and has a very sound software design.
Many of the client use-cases such as git clone have been implemented
According the answer to my question on the issue tracker [1]. The author also plans to implement parts of the server side stuff to allow you to build a server with it.
I guess it depends on what you need, but there's a few related projects out there.
The most "robust" implementation I can think of is this one by the 280North crew (of Cappuccino fame).
There's also some server-side JavaScript projects underway (e.g., http://github.com/ajaxorg/node-github), but that won't run entirely within a browser client.
Update (for anyone else who comes across this):
Please be sure to check out vanthome's answer. Tim Caswell's js-git project is well funded and undoubtedly the best answer here at this time.