Recs.
Updated
SpecsUpdate
Pros
Pro Free, Open Source & Permissive License
libGDX uses the Apache License 2.0.
Not only libGDX is free and open source but also it's license gives you a lot of power over the engine. As long as you provide a copy of the license, give credit, do not hold devs liable and do not use libGDX logo in any engine forks you can do pretty much anything you want.
Pro Active and helpful community
The libGDX community, in the official libGDX forum is extremely helpful and approachable for any kind of question regardless of the its quality or difficulty.
The forums themselves are a very helpful resource for any issue or guide simply by searching past posts in there.
In addition to the forums, there's also the official #libgdx IRC channel on Freenode.
Pro Fully flexible
You can do whatever you want, libGDX does not impose a way your game should be structured. You can include any library and it will work. There are no components, entities, nodes or scenes. This is great if you want it, at the cost of some time and the risk of creating a mess.
Pro Free, open source & permissive license
libGDX uses the Apache License 2.0.
Not only is libGDX free and open source but also it's license gives you a lot of power over the engine. As long as you provide a copy of the license, give credit, do not hold devs liable and do not use libGDX logo in any engine forks you can do pretty much anything you want.
Pro Focusing on object pool patterns, to control memory without pointers
Unlike Unity or other engine, it allows to optimize a language that uses garbage collector when using patterns of objects you can control the use of memory without needing a language like C / C ++, getting the same speed in a more productive language.
Pro Fantastic UI toolkit (scene2d)
scene2d is a 2D scene graph, scene2d.ui is a UI toolkit on top of that. It enables very complex mobile and desktop UIs.
Cons
Con Rather a framework than a game engine
LibGDX offers a lot of functionality, but it will usually take a lot longer to get things done, compared to a real game engine. This is because you have to program way more and because you usually include other libraries, that might not be connected in the best way.
Con Easy to make a bad structure for unexperienced programmers
If you do not have a lot of experience building and structuring larger programs, it is easy to get into a unworkable state. Since you are programming everything and there is no structure imposed, you have to create a good structure yourself, which can be difficult if you have little experience in that area. For example, there is no Scene System and there are no entities/nodes, you have to create this yourself.
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Out of Date Pros + Cons
Con Documentation could be better
Other than a brief installation / getting started overview, libGDX's documentation consists of an official wiki with several incomplete pages, and automated Javadocs. The community recognizes these shortcomings, and new users are encouraged to ask for help.