There seems to be service disruption on a few domains, so you will have to try later when it is resolved.

• http://status.mozilla.com/
• https://ftp.mozilla.org/pub/mozilla.org/

The https protocol is used to make it possible to use server load balancing.
The FTP server can only accept a limited amount connections and the https protocol makes it possible to redirect to other servers for load balancing.

It seems to be working at the moment (just updated Aurora and Nightly).

Thanks! I was now able to download the latest manually (took ~10 seconds), but the "Software Update" is still failing - I restarted and I've been looking at it for a few minutes by the time I finished typing the following...

It just says "Downloading Aurora" and shows a progress bar with "Connecting to the update server..." under the progress bar. I'll manually re-install and see if that solves my problem. I have noticed it failing for a while now and looking at "About Aurora" shows I'm on a 9-01-2013 version, but I forget how often they have been pushing Aurora.

I guess I'll assume the manual re-install fixed the update issue until I see otherwise.

Did you try to close and restart Aurora?

If something goes wrong with the updater then it can get stuck in connecting, especially if already a part has been downloaded (updates\0 folder contains an update.mar file) and in such a case you may have to remove updates\0 folder to make Firefox restart downloading the update.