The Vlasov-Maxwell equations are the equations of motion for a collisionless plasma: a high temperature, low density ionized gas in which electromagnetic forces dominate collisional effects. In this lecture we will probe the major open question: does the initialvalue problem have a smooth global solution for smooth data of unrestricted size in 3 space dimensions? That is, are there shocks in collisionless plasmas? Partial answers and known results will be surveyed, including weak solutions, solutions with small data, the resolution of the large-data problem in two space dimensions, classical vs. relativistic formulations and the use of symmetry.