Surface waves in a viscous incompressible fluid with surface tension in an inclined open channel of arbitrary cross section were studied. Both linearized and nonlinear problems are considered. An abstract approach based upon the theory of Hilbert spaces is adopted to investigate the linearized problem, and a perturbation method is used to study the nonlinear problem. The unique solvability of the linearized Navier-Stokes equations is proved by using the Galerkin's method. Centering around the concept of a critical speed of the surface wave, a singular perturbation scheme within the framework of long wave approximation is developed to study the asymptotic behavior of nonlinear surface waves in the channel. A critical Reynolds number is then obtained from this scheme as a criterion for the stability of the wave motion. By the use of various (L sub 2)-estimates, a rigorous justification of the singular perturbation is made for the linearized problem in which the channel surface is an inclined plane. (Modified author abstract)