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It is generally believed that the structure of spacetime as a classical manifold will cease to exist at distances of the order of the Planck length. A theory of gravitation on a noncommutative spacetime could be compatible with quantum mechanics and capture the expected nonlocality of physics at very small distances, while reproducing Einstein's general relativity at long distances. We consider gravitation as a gauge theory of the Poincar symmetry and aim at generalizing this point of view to noncommutative spacetimes. The important role of the Poincar symmetry in relativistic physics is reviewed. The formulation of quantum field and gauge theories on noncommutative spacetime is presented, emphasizing the recently discovered twisted Poincar symmetry of these theories. Some formulations of gravitation on noncommutative spacetimes proposed in the literature are also discussed. Finally, we study the possibility to formulate the twisted Poincar symmetry as a gauge symmetry of gravitational theory on a noncommutative spacetime. This book is intended for researchers and students who are interested in gravitation and elementary particle physics.
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