We consider augmented variational formulations for solving the static or time-harmonic Maxwell equations. For that, a term is added to the usual H (curl) conforming formulations. It consists of a (weighted) L2 scalar product between the divergence of the EM and the divergence of test fields. In this respect, the methods we present are H (curl, div) conforming. We also build mixed, augmented variational formulations, with either one or two Lagrange multipliers, to dualize the equation on the divergence and, when applicable, the relation on the tangential or normal trace of the field. It is proven that one can derive formulations, which are equivalent to the original static or time-harmonic Maxwell equations. In the latter case, spurious modes are automatically excluded. Numerical analysis and experiments will be presented in the forthcoming paper [Augmented formulations for solving Maxwell equations: numerical analysis and experiments, in preparation].