We present both an improved model and new experimental data concerning the problem of melting in multilayer adsorbed films. The model treats in a mutually consistent manner all interfaces in a stratified film. This results in the prediction of substrate freezing, a phenomenon thermodynamically analogous to surface melting. We also compare the free energies of stratified films to those of homogeneous films. This leads to an orderly classification of multilayer phase diagrams in the vicinity of the bulk triple point. The results of the model are compared with the experimentally known systems. Of these, only methane/graphite exhibits melting from homogeneous solid to homogeneous liquid in multilayer films. The systems Ne/graphite and Ar/graphite, studied by Zhu and Dash, exhibit surface melting and substrate freezing instead. We observe experimentally, by means of pulsed nuclear magnetic resonance, that melting in methane adsorbed on graphite extends below the film thickness at which the latent heat of melting is known to vanish. The multilayer melting curve in this system is a first-order prewetting transition, extending from triple-point dewetting at bulk coexistence down to a critical point where the latent heat vanishes at about four layers, and apparently extending to thinner films as a higher-order, two-dimensional phase transition. It would therefore seem that methane/graphite is an ideal system in which to study the evolution of melting from two dimensions to three dimensions.