In this introductory-level talk, I will present the main work of my thesis: the question of stability of the Vlasov-Maxwell System. This system, which describes the time evolution of plasma, is of significant interest for physicists, both since much of the universe is made up of plasma, and since there are many industrial applications involving plasmas. Moreover, the question of stability is of special interest to those attempting to construct fusion reactors. After exploring the basic mathematical and physical ideas, we will find a simple criterion that guarantees linear instability, and we will give concrete examples where it is satisfied.