Godot is free (libre), open source, MIT license, no royalties, 2D, 3D game engine full-featured. You can make any kind of complex games.
Pro Overall very easy to use
Granted, some buttons and features may be a little hard to find, but that is a problem that is present in almost all feature-packed software, and Godot engine is far from the hardest to understand game engine.
The user interface feels soft to the eyes, and many things can be intuitively figured out. The developers are also working hard to overhaul the API, and make it even more user friendly, with visual scripting planed for the near future.
Pro Unified game editor interface
All the game developing work is done inside one program: the engine editor. This feature is something only high-end engines have. Even the scripting is done in the same program. No need for Eclipse or other front-end editors.
Scene management, animation, scripting, and much more can be done in the same application, so you need only media editors and Godot to make games.
Pro Easy to learn scripting language
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 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.
Pro Friendly towards Version Control Systems
The engine is build 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 Internationalization of the editor
You can change the language shown in menus. Godot translations: https://hosted.weblate.org/projects/godot-engine/godot/
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.
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 Primarily supports own proprietary language (GD Script)
While it's very accessible, and if you know Python you'll pick it up fast, having to learn a new language to fully make use of the platform can be a bit discouraging. And for those learning to code as well as learning Godot for the first time, many would rather learn a language they can 'take with them' when they explore other platforms in the future.
Con Strange terminology at its base
Scenes can be made up of other scenes. That makes some sense. But even the smallest object (or prefab or asset) in a scene -- such as that spoon on the table or the marble on the floor -- is still called a scene... except when it's called a node. This is a bit odd for those coming from other engines. With all the great decisions behind the basic design of this engine, the choice of this term from all the potential other terms out there seems really out of place and only serves as a constant reminder that not everything about Godot is great.
Con Won't work on very old computers
Computers with OpenGL inferior to 3.3 won't be supported anymore on Godot 3.0 and superior, this means all computers that are about more than 10 years old.
While this may not seem that big of a deal for a 3D engine the 2D counterpart can work fairly easily on older computers without sweating a fret and is so almost exclusively a maintenance issue that could be altogether avoided.
Flagged Pros + Cons
Out of Date Pros + Cons
Con Not so good 3D game engine
Although the 3D game engine in Godot provides marvelous tools, it is incapable of producing reflection materials, or proper metal PBR. The engine also tends to get bloated incredibly easily.
*note: the rendering engine is supposed to be rewritten some time in the future to use Vulkan and fix these issues.
Con Slow built-in physics
In 3D, or through a GD scipt, physics run quite slowly in Godot (have a considerable impact on frame rate). However, the devs are working on the big 3.0 update for 3D, which should be finished in a month or two, and you can always code in C++ in Godot (if you know how). C# is also being implemented.