We present two new numerical techniques for solving electromagnetic scattering problems using integral equation-based methods. In the case of perfect electric conductors, we have developed a new formulation of the Maxwell equations in terms of well-posed boundary value problems for the vector and scalar potentials. This formulation permits the development of a well-conditioned second-kind Fredholm integral equation that is immune to the difficulties that plague standard approaches: spurious resonances, low-frequency breakdown, and sensitivity to the genus of the scatterer. In the case of penetrable inhomogeneous scatterers, we reformulate the electromagnetic scattering problem using a new constraint-free vector Helmholtz-like partial differential equation that is equivalent to the Maxwell equations. This also leads naturally to a well-conditioned Fredholm integral equation.