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A game engine is a software framework designed for the creation and development of video games. Developers use them to create games for consoles, mobile devices and personal computers.
For the purpose of this question, a 3D game engine is defined as a framework for managing game state and behavior, integrated with 3D-specific middleware (at minimum, a rendering engine). This is based on Wikipedia's definition of a game engine, specialized to 3D while taking note of the second paragraph: A game engine is the software, i.e. some algorithms, solely responsible for the game mechanics and is strictly speaking not the rendering engine.