Recently, there has been a dramatic increase in the use of high order finite element methods for the approximation of Maxwell’s equations. We shall discuss some of our own work in this area. In particular, we shall present families of hierarchic basis functions for the Galerkin discretisation of the space H(curl; Ω) that naturally arises in the variational formulation of Maxwell equations. The conditioning and dispersive behaviour of the elements is discussed along with approximation theory. Numerical examples are shown which demonstrate the accuracy and efficiency of the methods for computing solutions of the time-harmonic Maxwell’s equations.