This paper deals with reduced basis methods for the harmonic Maxwell’s equations. The authors examine several essential ingredients such as the a posteriori error estimates for the solution and output, off-line/on-line computation procedure, and two different greedy algorithms to build the reduced basis spaces. The method is then applied to a challenging electromagnetic cavity problem. Exponential convergence of the reduced basis approximation to the truth finite element approximation is observed. In the final part of this paper it is illustrated the performance of the algorithm in the framework of a non-trivial test case.