Do .net applications run on linux?
Are there any free/paid interop libraries available ?
The Mono project provides a standards compliant implementation of the CLR virtual machine component of .Net. They've also reverse engineered a significant number of the framework portion of .Net. You'll have significant issues trying to develop WinForms apps. Mono provides a list of several graphical toolkits you can use: http://www.mono-project.com/Gui_Toolkits (it looks like they actually support WinForms now; though I'm not sure of the extent of that support).
Note that the Mono port of Silverlight, Moonlight, is officially endorsed by Microsoft. So if you can get away with using that, it might be your best shot for cross-platform compatibility.
Note that you have the dotGNU project. It is an implementation of the .NET for linux.
They are not as feature complete as Mono. But it is worth mentioning.
Mono is a .NET-compatible platform, including compiler and runtime. The Mono Migration Analyzer helps figure out compatibility issues.
10 years after the question was first posted here! Now you can run .Net on Linux and iOS. The new generation of .Net is called dotnet core and is the future of the framework. https://docs.microsoft.com/en-us/dotnet/core/
Yes, with some caveats. It's called Mono.