[Here][1] is an SO question quite similar to this. However, through my own research, I haven't been able to find anything specific to .NET, but rather to just C#. 

You can install the runtime using APT on a Debian distro by executing:

    $ sudo apt-get install mono-runtime

You can also (assuming you have some sort of GUI such as LXDE) install a *slow* Mono IDE with:

    $ sudo apt-get install monodevelop

For Arch Linux ARM you need to install the runtime via pacman, like so.

    $ sudo pacman -S mono

The Mono IDE can be installed in a similar fashion.

    $ sudo pacman -S monodevelop

  [1]: http://stackoverflow.com/questions/10213600/mono-on-raspberri-pi