This paper treats the problem of the refraction of a shock wave at a gaseous interface. The governing equations are formulated and analyzed. Continuous families of solutions are obtained numerically for a number of gas combinations at varying angles of incidence on a plane interface between ideal gases (characterized by a certain range of parameters). It is believed that these solutions, which represent a three-wave configuration at the interface with a reflected shock wave or with a reflected rarefaction wave, are physically real inasmuch as they tie in with the two known limiting solutions of an infinitesimal shock at any angle of incidence and of any finite shock at normal incidence. Two of the significant features of the present solutions are: (1) regular refraction (three-wave configuration at the interface) does not occur at glancing incidence and (2) in the region of regular refraction there is no "total reflection" of finite shock waves.