Poetically, “Group theory is the branch of mathematics that answers the question “What is symmetry?” (N. C. Carter).  Various physical systems, such as crystals and the hydrogen atom, can be modeled by symmetry groups. Group theory and the closely related representation theory have many applications in physics and chemistry. We first start by consider some main classes of groups such as permutation groups, matrix groups, transformation groups, topological and algebraic groups. We will study representation of groups and symmetry and the last part of the course is devoted to some applications of group theory such as Lie groups and the study of differential equations, and the Heisenberg, Lorentz, and Poincare groups, which play a fundamental role in modern theoretical physics.