We consider the possibility of superconductivity in a system of two-dimensional electrons on the surface of liquid helium films. Taking into account of the interaction between electrons and the surface excitations of liquid helium films-ripplons, within the weak coupling BCS theory, we estimate the superconducting transition temperature for various interaction strengths, film thicknesses, and electron densities. The superconducting transition temperature Tc, under experimentally realizable conditions, is calculated to be a few mK's. © 1993.