BDX
BDX is a small 3D game engine, designed to work with Blender (as an addon), and built to leverage libGDX, along with the Bullet physics library (specifically, the gbullet port). It features a straightforward execution model, supported by a relatively simple API, enabling the programmer to effectively define object behavior within a typical blender scene.
From a "build and deploy" perspective, BDX projects are simply extended libGDX projects, which means that you can use the same gradle-powered pipeline to deploy BDX games to any platform that's currently supported by libGDX (Windows, Linux, Mac OS X, HTML5, Android and iOS).
License
BDX is licensed under the permissive Apache 2 License (just like libGDX): You can use this software free of charge, no strings attached, in commercial and non-commercial projects.
Dependencies
For game development with BDX, you only need a relatively recent version of the Java Development Kit (I tested with 1.7).
If you plan on developing the engine itself, you'll additionally need a working version of ant.
Download and Install
You can find the latest release here.
Download bdx.zip. Once you have it, you can install it like any other blender addon - In User Preferences, click on the "Install from File" button, then find and install bdx.zip.
After that, you just need to enable it (you can find it in the Import-Export category - Testing support level), and then Save User Settings, so you don't have to re-enable after every blender restart.
At that point, there should be a BDX panel in the Render properties window.
Use
You can find relevant information in the wiki.