Abstract. This will be an elementary introduction to the Galois theory of linear differential equation. I will explain, from both the analytic and algebraic points of view, how one associates a group of matrices to a linear differential equation. I will discuss how this group is useful in determining when one can solve a linear differential equation in terms of exponentials, integrals and algebraic functions and also how this group can be used to give necessary conditions for a Hamiltonian system to be completely integrable. Finally, I shall discuss algorithms (and implementations) for calculating this group and its properties.