Solutions to boundary values problems for Maxwell’s equations are described using the layer potential Maxwell operator. The Maxwell, electric and magnetic boundary value problems for time harmonic electromagnetic wave propagation have unique solutions in C ¹ domains and satisfy optimal estimates. Sobolev-Besov regularity results hold for domains with Lipschitz boundaries. Also, multipole expansion methods can be used to approximate L ^p tangential vector fields on the boundaries of domains which are only Lipschitz.