In recent years, a new field integrating robotics and topology was born, referred to by many as Topological Robotics, in which the main strategy is to use algebraic topological tools to get some insight into robotics problems. One of those problems is called the robot motion planning problem and is the main motivation for this work. We present an in-depth study of Topological Complexity, discussing how it relates to the Motion Planning Problem, and the main methods for computing it for CW complexes and Smooth Manifolds, spaces of great interest in robotics. The concept of Lusternik- Schnirelmann category is introduced due to its connection with Topological complexity, both being particular cases of the more general concept of the Schwarz Genus of a fibration.

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