In this chapter, we consider the nonstationary Maxwell system in a simply connected open cone in ℝ³ with boundary smooth outside the vertex, in a wedge, and in a bounded domain in ℝ³ with a conical point. The Maxwell system is endowed with the boundary conditions which correspond to the case of perfectly conducting boundary. For studying this system there will be used a scheme based on a proper combined estimate. We describe the asymptotics of solutions near singularities of the boundary.