Measurements of the effective thermal conductance of a He4 film on a Mylar substrate are reported in the vicinity of the transition to superfluid behavior. Experiments have been conducted as a function of film thickness and are consistent with the expected exponential behavior of the free vortex number density as the transition is approached from the normal-fluid side. On the superfluid side of the transition the effective conductance is observed to be a power law in the applied thermal power. The connection between the exponent of the power law and the superfluid density is examined. The experiments reveal an apparent upper limit to the number density of velocity-induced free vortices. Preliminary results for He3-He4 mixtures are reported as are nonlinear effects in the He4-vapor conductance.