In Mechanics, one may use various geometric arguments to reduce the total number of equations needed to describe a system. In particular if the configuration space of the system is a Lie group, then one can use the group structure to simplify the governing equations. We will develop some of the theory of mechanics on Lie groups, paying particular attention to the case of rigid body dynamics, and showing how this leads naturally to the Euler-Poincare equations of motion. We shall then show how these equations can be used to obtain a new Lagrangian derivation of the Maxwell-Vlasov equations of plasma physics.