Caveolae are cell membrane invaginations of defined lipid and protein composition that flatten with increasing membrane tension. Super-resolution light microscopy and electron microscopy have revealed that caveolae can take a variety of cuplike shapes. We show here that, for the range in membrane tension relevant for cell membranes, the competition between membrane tension and membrane bending yields caveolae with cuplike shapes similar to those observed experimentally. We find that the caveola shape and its sensitivity to changes in membrane tension can depend strongly on the caveola spontaneous curvature and on the size of caveola domains. Our results suggest that heterogeneity in caveola shape produces a staggered response of caveolae to mechanical perturbations of the cell membrane, which may facilitate regulation of membrane tension over the wide range of scales thought to be relevant for cell membranes.