MATE is not currently available in the official Ubuntu / Debian repositories but it seems it will make it soon because it's currently on Debian ftp-master new queue.
Ubuntu 12.10, 12.04 and 11.10 users can install MATE Desktop by using its official repositories. To add the repositories and install MATE, use the following commands:
[sourcecode language="bash"]
sudo add-apt-repository "deb http://packages.mate-desktop.org/repo/ubuntu $(lsb_release -cs) main"
sudo add-apt-repository "deb http://repo.mate-desktop.org/ubuntu $(lsb_release -cs) main"
sudo add-apt-repository "deb http://mirror1.mate-desktop.org/ubuntu $(lsb_release -cs) main"
sudo apt-get update
sudo apt-get install mate-archive-keyring
sudo apt-get update
sudo apt-get install mate-core mate-desktop-environment[/sourcecode]
Then, log out and select MATE from the login screen.
Note: for some reason, in my test, the MATE Notification Daemon wasn't installed by default so if that's the case for you too, use the following command to install it:
[sourcecode language="bash"]sudo apt-get install mate-notification-daemon[/sourcecode]
I've also seen some Raring packages in the MATE repositories so it seems Ubuntu 13.04 Raring Ringtail will be supported soon.
For other Linux distributions, see the MATE installation wiki page.
Removing MATE
If you've installed MATE in Ubuntu using the commands above and you want to remove it, use the command below in a terminal:
[sourcecode language="bash"]sudo apt-get remove mate-archive-keyring mate-notification-daemon atril atril-common caja caja-common engrampa engrampa-common eom eom-common gir1.2-mate-panel libatril libcaja-extension libmarco libmatedesktop libmatekbd libmatekeyring libmatemenu libmatepanelapplet libmatepolkit libmateweather libmateweather-common libmatewnck libmatewnck-common marco marco-common mate-applets mate-applets-common mate-backgrounds mate-calc mate-control-center mate-core mate-desktop mate-desktop-common mate-desktop-environment mate-dialogs mate-icon-theme mate-media mate-media-common mate-media-gstreamer mate-menus mate-panel mate-panel-common mate-polkit mate-power-manager mate-power-manager-common mate-screensaver mate-screensaver-common mate-session-manager mate-settings-daemon mate-settings-daemon-common mate-settings-daemon-gstreamer mate-system-monitor mate-terminal mate-terminal-common mate-themes mate-utils mate-utils-common pluma pluma-common[/sourcecode]
And to remove the MATE repositories, use:
[sourcecode language="bash"]sudo add-apt-repository -r "deb http://packages.mate-desktop.org/repo/ubuntu $(lsb_release -cs) main"
sudo add-apt-repository -r "deb http://repo.mate-desktop.org/ubuntu $(lsb_release -cs) main"
sudo add-apt-repository -r "deb http://mirror1.mate-desktop.org/ubuntu $(lsb_release -cs) main"
sudo apt-get update
[/sourcecode]
Happy using MATE!