We develop a physical theory for the light curves of contact binaries based on the assumption that the dynamical time scale is very short in comparison with the thermal time scale at the base of the common envelope. In contrast with the case for common radiative envelopes the flux distribution in common convective envelopes does not exhibit any effect of gravity brightening. Combined with a unified treatment of reflection and limb darkening, this result produces W-type light curves for W UMa stars of special type later than F5 if the orbit is inclined by less than approximately 70°-90°. The sign of the effect is in rough accord with the observations, but some discrepancy remains concerning the magnitude of the effect. We speculate that the interaction between rapid rotation and convection may contribute to the remaining discrepancy; it may also produce asymmetry and time variability which are observed in some light curves.