The results of an experimental study using torsional oscillators of the superfluidity of4Hefilms adsorbed on hydrogen plated graphite are reported. The evolution of superfluidity with the growth of the film shows considerable structure arising from the atomic layering of the film. There is evidence that the superfluidity of a single fluid layer is strongly suppressed, possibly due to the influence of the periodic potential arising from the underlying solid layer. The behavior of two fluid layers is quite distinct, but shows similarities to that of thicker films on heterogeneous substrates; we suggest that the usual theory of the superfluidity of two-dimensional4Heshould be extended to account for superfluid onset temperatures in such a film. ©1997 American Institute of Physics.