We investigate the adhesion of vesicles, under the influence of a contact potential, to substrates with various geometry. For axisymmetric configurations, we find that the transition from a free vesicle to a bound state depends significantly on the substrate shape. In general, the critical values of the contact potential at which these transitions take place are lower for a concave-shaped substrate than that for a flat-shaped substrate investigated in earlier studies. We observe that the transitions happen at higher critical values of the contact potential when the substrate is convex and illustrate how these critical values depend on the curvature of the substrate. In addition, we construct an approximate analytical solution that predicts the shape of the vesicle for large internal excess pressure and contact potential. The analytical solution leads to an inequality that relates the surface tension with the contact potential. © 2008 The American Physical Society.