Open source edition: Apache v2
(Closed source edition also exists)
LinuxOnAndroid, also known as Complete Linux Installer is an application for Android devices that allows users to install an X Window based Linux distribution, such as Ubuntu or Debian, enabling them to use Linux desktop applications in Android. The app has been downloaded between 100,000 - 500,000 times.
The Linux install is on a loopmounted image file, however, is able to access the Android filesystem. VNC is used to display the GUI of the X Server. It requires root access of the phone.
The authors are planning to release a ROM similar to Ubuntu for Android, which will integrate the two operating systems. They had crowdfunded for this purpose using the Indiegogo website.