An asymptotic theory is developed for the study of three‐dimensional nonlinear surface waves on an incompressible, viscous fluid with surface tension in an inclined channel of arbitrary cross section. The method used here is based upon a multiparameter singular perturbation scheme within the framework of long‐wave approximation. The nonlinear problem is reduced to a sequence of linear elliptic mixed boundary‐value problems, which may be solved by means of known methods. Their solutions are then used to determine the wave speed and evolution equations governing the nonlinear wave motion. The results obtained give a quantitative description of a three‐dimensinnal bore structure in an inclined channel of arbitrary cross section, and a critical Reynolds number is also defined as a criterion for the instability of the wave motion.