The objective of this talk is to report some of our recent efforts in exploring the possibility of extending the general framework of adaptive finite element methods based on {/em a posteriori} error estimates to resolve Maxwell singularities. {/em A posteriori} error estimates are computable quantities in terms of the discrete solution and known data that
measure the actual discrete errors without the knowledge of exact solutions. They are essential in designing algorithms for mesh modification which equi-distribute the computational effort and
optimize the computation. We will consider the time-harmonic cavity problem, the transient eddy current problem, and the electromagnetic scattering problems.