The effect of an electric field on a liquid layer flowing down an inclined, corrugated wall at zero Reynolds number is investigated. The layer is taken to be either a perfect conductor or a perfect dielectric. The region above the layer is assumed to be a perfect dielectric. Steady flow down a wall with small-amplitude sinusoidal corrugations is considered, and it is shown how the electric field can be used to control the amplitude of the free-surface deflection and the phase shift between the free surface and the wall profile. Steady flow over walls with large amplitude sinusoidal corrugations or other-shaped indentations is studied by using the boundary-element method. Results for flow into a wide rectangular trench are compared to previous model predictions based on the lubrication approximation. For a perfect-conductor film, the results confirm that the height of the capillary ridge, which appears above a downward step, monotonically decreases as the electric field strength increases. Solutions for a perfect-dielectric film with relative permittivity larger than unity are similar to those for a perfect-conductor film, although the height of the capillary ridge nonmonotonically varies with the electric field strength. The behavior of the solutions for a perfect-dielectric film with relative permittivity less than unity is qualitatively different. The height of the capillary ridge monotonically increases as the electric field strength increases. Flows into narrow trenches and over narrow mounds are also computed.