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Quantum gravity is an area of research in physics. Right now, physics can describe gravity of real-life-sized objects using general relativity. Physicists can also describe the universe on a microscopic scale with quantum mechanics. The goal of quantum gravity is to develop a theory that contains both general relativity and quantum mechanics. However, combining the theories of the big and the small is very difficult in practice.
Physicists are working on several ways to write down a theory of quantum gravity. One approach is to search for a unified field theory. Physicists also try to develop theories at an even more fundamental level, like string theory, loop quantum gravity, or other methods, to find out why our current theories of gravity and quantum mechanics cannot be combined.