LibGDX is an open-source cross-platform Java game development framework based on OpenGL (ES) that works on Windows, Linux, Mac OS X, Android, iOS and WebGL enabled browsers.
XNA is a cross-platform framework originally from Microsoft that facilitates game development with .NET
"Xna is more popular and has more tutorials then libgdx"
from question "LibGDX / Java or XNA / C#"
"Xna + monogame --- monogame s performance seems only slightly lower than libgdx can build to most platforms"
from question "2D Cross-Platform Game Development Engines"