Recs.
Updated
Godot is free(libre), open source, MIT license, no royalties, 2D, 3D game engine full-featured. You can make any kind of complex games.
SpecsUpdate
Pros
Pro Active development
This engine, barely released one year ago, has more than 1000 forks on github and about 100 developers. Not only that, just a bit of browsing through issues you will quickly find out the dev community loves new esp free technology and does not shy away from completely rewriting parts of the engine. The audio engine is being completely rewritten to use threads and so forth.
Pro Easy to learn scripting language
Godot has their own scripting language called GDscript. The scripting language is easy to learn with python-like format, but it is not python. More like a mix of Javascript, PHP, C++. It's very powerful, easy to learn, and it's free of unnecessary things because it's designed for this purpose.
It can be used to add custom behaviors to any object by extending it with scripting, using the built-in editor with syntax highlighting and code completion.
A built-in debugger with breakpoints and stepping can be used and graphs for possible bottlenecks can be checked.
Pro Really good community
The community is great and really cares about the engine. It is easy to get help and to be part of Godot's future.
Pro Friendly towards Version Control Systems
The engine is built not only to support version control but to really use it. Scene files, for example, which usually get compiled into some sort of unreadable data, stay in a text format - that way you can actually see your changes in a version control system like Git.
Pro Easily expanded scripting system
With 3.0's addition of NativeScript and PluginScript via GDNative, developers can easily define bindings for new scripting languages. In addition to the primarily supported C++, GDScript, VisualScript, and C# languages, the community has contributed D, Nim, and Python as well with more on the way.
Pro Simple and readable codebase
The engine's source code is easy to read and understand with a self-documenting approach to code design. You don't have to wait months or years for other people to fix an engine bug that is important to your game. Often times, you can spend an hour or two of your own time to fix whatever problems you encounter yourself.
Pro Built-in documentation linked to the internal ScriptEditor
The editor has a fully searchable index of class API documentation for everything the engine offers (NOT just a web interface). You can easily open the documentation for any class by Ctrl-clicking the class's name in the in-engine text editor for scripts.
Cons
Con Not as beginner-friendly as all the fanboys claim it to be
Not as beginner-friendly as all the fanboys claim it to be. The interface is confusing. Dark mode only appeals to wannabe hackers.
Con Difficult to optimize
Godot has an OOP architecture. Everything is an object internally and data is spread among many classes, thus it's difficult to optimize (i.e. not cache friendly, difficuly to vectorize or paralellize, etc).
Read about "Data Oriented Design" for more info about the problems and solutions.
Con No console targets
Given that you can target both desktops and consoles with the same code base in other engines, the lack of support for consoles in Godot is pretty hard to get past if targeting desktops for a game. But asking for an open-source engine to target consoles is probably too much to ask. But it would be interesting to see some legacy consoles targeted even if current ones cannot be.
Con Very bad documentation
The documentation is poorly written, and has very few examples of real application and even fewer design guidelines about how to program a game in the engine.
Con No native Admob or other AdNetwork support
Godot has no native support for Ads but there are functional plugins and modules that do that.
Con NativeScript is limited to what GDScript can do
All language bindings are extensions for use with NativeScript, which means that even if you used something like Rust, you could only do what GDScript can. You will be required to use C++ for any low-level control over the engine in your games.
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Flagged Pros + Cons
Pro Scene Based editing
Godot gives you the ability to create scenes to make your life easier, with reusable objects and things you want to incorporate in your games. This makes the game making processvery streamlined and organized.
Pro Multi-platform deploy
Deploy games on desktop (Windows/OS X/Linux), mobile (iOS/Android/BlackBerry), and web (HTML5 via Emscripten).
Pro Under constant development
This engine, barely released one year ago, has more than 1000 forks on github and about 100 developers. Not only that, just a bit of browsing trough issues you will quickly find out the dev community loves new esp free technology and does not shy away from completely rewriting parts of the engine. The audio engine is being completely rewritten to use threads and so forth.