A while ago Somasegar's blog post about MS not supporting Visual Studio 2003 on Windows Vista caused quite a stir in community. A lot of people were very concerned about this issue.
A note at this point: .net 1.1 and newer will be fully supported. It is just the Visual Studio that has problems. While they are putting all their effort in Visual Studio 2005 (through Service Pack 1 - it will be required - and there might still be some minor issues) and Visual Basic 6 (apparently still alive) they are leaving Visual Studio 2003 behind. The suggestion here is to use virtual machine to run Visual Studio 2003 on some other OS within this virtual environment. However, this is not an ideal solution for several reasons - the most important being that not everything runs in virtual machine environment, such as DirectX applications. The other almost equally important is slowness of virtual machines.
Back to the cause of the problems. Most of the problems are caused by Vista User Account Control (UAC) which is very aggressive in protecting the OS from various threats. And the side effect is that it nukes debuggers. In other words, you can run Visual Studio 2003 but you can't debug (remember old times, when we were putting a ton of "Output" methods in code instead of debugging).
Here are the good and the bad news: It is possible to run Visual Studio 2003 including debugger (at least in normal debugging situations - debugging WinForms, Console applications - I didn't tested others) in Vista but it comes with a price. You have to turn off UAC (you can do it by running msconfig utility) and switching on/off UAC requires a machine restart. Yep, I have turned it off because I need Visual Studio 2003 for DirectX managed applications. I guess I won't be using UAC for a while.
In my opinion (and in opinion of many others) MS' decision to support VB6 instead of VS2003 remains very questionable.
UPDATE (23.12.2006): Dan posted a comment with link to "what's not working in vista and how to workaround it".