How do I develop .NET apps on Ubuntu?

You can program in .NET on ubuntu too. Well, sort of. There is an open source implementation of the .NET platform available called MONO. MONO apps can run on Ubuntu/Linux, Windows and Mac OS. Look for MonoDevelop in Ubuntu Software Center. Learn more about Mono in Ubuntu.

Another option is Quickly. In my opinion Quickly is better for Ubuntu centric app development. You can code you app and release it to a PPA (launchpad-repository) in minutes. Quickly is also available from Ubuntu Software Center. Get started writing apps with Quickly.

Quickly can help you make cool apps using Python (pretty easy to learn and very popular) quickly!

The Vala language is a programming language very similar to C#. It is still very young, but already has a lot of bindings to existing libraries (for example, GTK). It compiles to native binaries, so your users don't need an additional runtime, as they would for .NET.

See an introduction to Vala for C# programmers.