How do I use Visual Studio to develop applications on Mono? Is this possible?
You just build the applications in Visual Studio, and run them under Mono instead of under .NET. The binaries should be compatible... but you'll need to make sure you don't use any libraries which aren't available in Mono - see the Mono Application Compatibility Guidelines.
(According to that page, you need to turn off incremental builds in Visual Studio though - a point I wasn't aware of before :)
Miguel has posted this entry a while ago, so it's quite dated. You can also try this.
Since version 2.0 MonoDevelop supports VisualStudio project/solutions file format. This means that you can use the same code base on Windows with VS and .Net and on Linux with MonoDevelop and Mono. That in my opinion is the best way to go about it. There are no major reasons to run application in Mono on Windows, other than for testing purposes and for that I'd rather set up a virtual machine to test the software in native environment.
I've posted an article on how to integrate Mono 2.8 (the build for .Net 4.0) into Visual Studio 2010 here - there's a link there to another profile built for Mono 2.4 (.Net 3.5) if that's what you're aiming for.