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The gyrotron (or, electron cyclotron maser) operates on the interaction of a TE cavity mode with the relativistic electron cyclotron beam mode. This instability has been seen both theoretically and experimentally to be a very efficient source for microwave generation, useful for the heating of tokamak plasmas at the electron cyclotron frequency. The next generation of tokamaks involves high confining fields and accordingly the gyrotron must produce short (approx. 2mm) wavelength radiation, which can be realized only by a highly overmoded operation of the gyrotron. This work presents the groundwork for the study of such an overmoded operation. The self-consistent nonlinear equations have been obtained for the evolution of the electron distribution and of the power spectrum for multimode gyrotron interactions in a rectangular cavity or waveguide. The time step of the resulting algorithm is limited only by the nonlinearities and nonuniformities, and therefore yields very efficient simulations. The equations have been applied to the following cases: (a) in a waveguide, analytically, to obtain the dispersion relation and saturation behavior of a monochromatic wave with two-dimensional transverse structure; (b) in a waveguide, numerically, to test the correctness and accuracy of the computer code; (c) in a waveguide with external field (or equivalent) nonuniformities, to interpret the associated efficiency enhancement; and (d) in a two-mode cavity, to study the evolution and saturation of the eigenmodes. (Author)