The adhesion of fluid vesicles to a planar surface has been studied using Monte Carlo methods and scaling arguments for the random surface model. Deflated as well as inflated vesicles have been considered. Inflated vesicles, with internal pressure p>0, exhibit with increasing adhesion strength a discontinuous conformational transition from unbound sphere-like conformations to strongly adsorbed two dimensional branched conformations. Deflated vesicles with p=0 exhibit a continuous transition from three dimensional to two dimensional branched conformations where the type of transition is different from adsorption transition of branched polymers. The transition temperatures scale according to epsilon(c)/kT similar to p root N, where N is the surface area of the vesicle. (C) 1997 American Institute of Physics.