There is a StackOverflow question quite similar to this, Mono on Raspberry Pi. However, through my own research, I haven't been able to find anything specific to .NET, but rather 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