We consider a vesicle carrying on its surface a small number of mobile "stickers". When facing a wall with suitable receptors for the stickers, the vesicle builds up an adhesion patch. Our aim is to analyze the patch growth, allowing for a variety of physical situations. (a) Sometimes the process is facilitated by a nonspecific adhesion process (e.g., van der Waals). But with suitably protected surfaces, this is absent: we can then use heavy vesicles, who fall by their weight on the surface but remain at a certain distance from it. N For certain model systems, the stickers diffuse freely. But for other systems, they may be stuck as soon as they reach their complement. (c) With some cellular adhesion molecules, we may find a long reaction time tau for fixation. Our discussion includes all these possibilities and ends up with a list of possible scenarios, leading to different growth laws.