No. The Microsoft .NET Framework is not platform independent. It only runs on Microsoft Windows (and therfore x86 and x64 processors).
However, there are open source implementation's of Microsoft's .NET Framework that are compliant with the ECMA standards for the C# programming language, the Common Language Runtime, and the Framework Class Library.
Novell's Mono project for example:
http://www.mono-project.com/Main_Page.
Mono runs on pretty much any operating system and/or processor architecture on the planet: Linux, Microsoft Windows, Mac OS X, BSD, Sun Solaris, Nintendo Wii, Sony PlayStation 3, Apple iPhone; x86, x86-64, IA64, PowerPC, SPARC (32), ARM, Alpha, s390, s390x (32 and 64 bits) and others.
The Microsoft .NET Framework's Common Language Runtime's JIT compiler generates x86 instructions (32-bit mode) and x86-64 instructions (64-bit mode).
I've poked around in the Mono source a bit, but have only programmed against the .NET framework running under Windows.