Disclaimer: This instructions are outdated.
visit https://github.com/ppy/osu#building-and-running for current building instructions
For those who wanna play or try osu! on Linux but don't want to use Wine there is a new version of the client currently in development called osu! Lazer and it's open source!
But first, as stated on the GitHub page of the project:

This is still heavily under development and is not intended for end-user use. This repository is intended for developer collaboration. You're welcome to try and use it but please do not submit bug reports without a patch. Please do not ask for help building or using this software.

This means altought you can log-in with your acount, there's no online match and you can expect a lot of inestability and bug.
So in order to try this we have to build it from source.
I'm using Linux Mint 18.1 so this tutorial will focus on Debian based distributions.

In order to build it you gonna need to install some tools:

• 
  
  
  • Wget to get some files from the web
  • Git to get the source code
  • Mono-develop to compile it
  • And the source code itself

First open your terminal

Spoiler: Git and Wget

You can get both from your distribution repository so just:

$ sudo apt install git
$ sudo apt install wget

Spoiler: Mono

Mono can also be found in your distribution repositories but it will be outdated so first we add the MonoDevelop Project repository:

for Ubuntu 16.04 (or distributions using the same kernel):

$ sudo apt-key adv --keyserver hkp://keyserver.ubuntu.com:80 --recv-keys 3FA7E0328081BFF6A14DA29AA6A19B38D3D831EF
$ echo "deb http://download.mono-project.com/repo/ubuntu vs-xenial main" | sudo tee /etc/apt/sources.list.d/mono-official-vs.list
$ sudo apt-get update

for Ubuntu 14.04 (or distributions using the same kernel):

$ sudo apt-key adv --keyserver hkp://keyserver.ubuntu.com:80 --recv-keys 3FA7E0328081BFF6A14DA29AA6A19B38D3D831EF
$ echo "deb http://download.mono-project.com/repo/ubuntu vs-trusty main" | sudo tee /etc/apt/sources.list.d/mono-official-vs.list
$ sudo apt-get update

Then simply:

$ sudo apt-get install monodevelop

These instruction alongside support for other distros and platforms can be found he: http://www.monodevelop.com/download/#fndtn-download-lin-ubuntu

Building osu!

Use Git to get the source code from github:

$ git clone --recurse-submodules https://github.com/ppy/osu

then CD into osu and use Git to download nuget.exe

$ cd osu
$ wget https://dist.nuget.org/win-x86-commandline/latest/nuget.exe

now restore the packages using nuget

$ mono ./nuget.exe restore

Spoiler: ignore this if the restore gives no errors

when restoring the nuget package i received warning about 'Detected package downgrade'
i solve the issue by Installing .NET Core SDK from Here

Finally use msbuild to 'build' osu! this will take some time and it should give 0 errors and 0 warnings.

$ msbuild

you can now run osu! using mono from ~/osu/osu.Desktop/bin/Debug/net471/osu.exe!

$ mono ~/osu/osu.Desktop/bin/Debug/net471/'osu!.exe'

 

[JellyPerson]

Nice! osu is a good game, and now that it's going to be on linux, it's going to be extra awesome

 

[orangy57]

I'm surprised that the game has been around for so long and yet none of the devs thought to port it to other systems