Debian

This app will install Debian Jessie.
You do not need to root your device, it will work fine with any stock ROM.
You will need 900 Mb free on internal storage. This app cannot be installed to SD card.
A mouse or stylus is highly recommended.
Android 6.0 Marshmallow is not supported yet (apt-get and dpkg do not work).

This app is NOT full Debian OS - it is a compatibility layer, which allows you to run Debian applications.
Your phone is NOT rooted during installation.
This is NOT official Debian.org release.

If you have Error -24 when installing this app - install this version (it's the same app with different name):
http://sourceforge.net/projects/libsdl-android/files/ubuntu/Debian-15.08.28-xx.apk
Do not uninstall this app if you have Android 4.4 or older - you will not be able to install it again, because of a bug in Android, which was fixed in Android 5.0.
Clear app data instead to save space - long-press app icon in drawer, drag it to App Info icon, and select Clear data.

If you want to use web browser, open Terminal and run command sudo apt-get update ; sudo apt-get install iceweasel

If you want to use Eclipse or other Java applications, install package openjdk-7-jdk and restart Debian app, because default JDK does not work on Samsung devices.

To disable gyroscope, tap Change device config during app start, select Mouse emulation → Advanced → Gyroscope.

To set 24bpp color depth or portrait screen orientation, go to Change device config → Video.

If your favorite music player does not work - install package gstreamer0.10-pulseaudio.

Sources are here:
https://github.com/pelya/commandergenius/tree/sdl_android/project/jni/application/xserver-debian
Previous versions are here:
https://sourceforge.net/projects/libsdl-android/files/ubuntu/

This app uses PRoot. All hail PRoot and it's most gracious developer, Cédric VINCENT!