The current latest version:  Kodi v16.1 (Jarvis)

Follow these in instructions to install Kodi on an Amazon Fire stick.

• Turn the Firestick on
• Go to Settings
• Select System
• Select Developer Options
• Select Apps from Unknown Sources and turn ON & Select ADB Debugging & turn ON
• A warning message will pop up, select OK 

• Next, you now need to install ES File Explorer. You will find this by searching in the Amazon App Store within the FIRE TV app menu.
• Once its installed, then open ES File Explorer.
• Go to Settings > Applications > Manage installed Applications > ES File Explorer then Launch it.
• Once opened, choose ‘Add‘.
• Then enter:

Path: http://www.gabbotech.com/kodi/k16_1.apk
Name: Kodi161

• Then click ‘Add‘ to save. This will add a bookmark to the menu on the left off the screen.
• Scroll down and select ‘Kodi161‘.

• It should now download Kodi and go though the install process.
• Follow the onscreen instructions, selecting the necessary choices on-screen to install.
• Kodi should now be installed