Oh....that's big topic to cover. It really depends on what it is you are wanting to do. Their are several services that allow for basic audio transfering. Digidesign has a studio transfer service, so did logic at one point - don't know if they still do. You have to sign up and pay for bandwidth etc. There are a few new sites up and running designed for large file transfering.
Of course you can just do it via your own server and FTPing.
The internet as a global studio is mostly limited by your bandwidth. If you have access to big pipes (T1 line) you can do more.
Transfering large files on your average DSL/Cable Modem at high speed is still very time consuming. It can be done, but even loading a large session on a DVD and sending express mail would be probably be faster and cheaper. Not to mention it won't tie down your computer for long periods of time. If I have only a few files to send, I'll upload them to my site for downloading. If I have a big session, I'll send it on a CD-R or DVD-R.
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What the hell is that thing?