The Lusternik-Schnirelmann category associates a positive integer to each topological space. This number is an important invariant in algebraic topology, critical point theory and symplectic geometry. In this dissertation we present the theory of Lusternik-Schnirelmann category, compute the category of several topological spaces and provide some reformulations of the category. In addition, we show two applications of Lusternik-Schnirelmann category to other areas of mathematics. The first one is a geometric application proving that a convex body in Euclidean space of dimension n admits at least n binormal chords. The second application relates the category to topological complexity in the motion planning problem.