This article describes Monte Carlo algorithms for charged systems using constrained updates for the electric field. The method is generalized to treat inhomogeneous dielectric media, electrolytes via the Poisson-Boltzmann equation and considers the problem of charge and current interpolation for off lattice models. We emphasize the differences between this algorithm and methods based on the electrostatic potential, calculated from the Poisson equation. (C) 2004 American Institute of Physics.