Beam matching is a common accelerator design and operation technique to minimize beam losses. Although widely used, a full theoretical understanding of beam matching in 6D remains elusive. In this paper, we present an analytical treatment of 6D beam matching of a high-intensity beam in RF structures. We start within the framework of a linear model and apply the averaging method to a set of 3D beam envelope equations. Accordingly, we obtain a matched solution that is comprised of smoothed envelopes and periodic terms, describing envelope oscillations with the period of the focusing structure. We then consider the nonlinear regime, where the beam size is comparable with the separatrix size. Starting with a Hamiltonian analysis in 6D phase space, we attain a self-consistent beam profile, showing that it significantly differs from the commonly used ellipsoidal shape. Subsequently, we analyze the special case of equilibrium with equal space charge depression between all degrees of freedom. A comparison of beam dynamics for equipartitioned, equal space charge depression, and equal emittances beams is given. Finally, we present experimental results on beam matching in the LANSCE linac.