Liquid droplets, gas bubbles, and membrane vesicles which are in contact with chemically structured substrate surfaces can undergo morphological transitions or shape transformations. The structured surfaces considered here consist of two types of surface domains, γ and δ, which attract and repel the droplets, bubbles, and vesicles, respectively. For droplets on a striped γ domain, one has to distinguish droplets with fixed end caps from those with freely moving end caps. Both types of channels undergo morphological wetting transitions. For vesicles, one has a strong adhesion regime in which the vesicle shapes have constant mean curvature and exhibit effective contact angles. One can then map the shape bifurcation diagram for vesicles onto the one for droplets if one includes the constraint of fixed membrane area. We also report preliminary experimental observations of the adhesion of vesicles to chemically structured surfaces.