<p>This paper presents a study on adhesive contact between an elastic half-space with a flat surface and a rigid half-space with a rectangular-grooved surface. The adhesion forces are taken into account by utilizing the Maugis–Dugdale model. The solutions are obtained for non-adhesive contact, adhesive contacts with and without contact zone inside the grooves by solving singular integral equations. The effects of the applied load and adhesive stress on the adhesive contact are studied for loading and unloading stages. The adhesion hysteresis, jump-in to contact and pull-off phenomena are revealed. The direct transition from the non-adhesive contact to the adhesive contact with contact zone occurs for intermediate and strong adhesive stress. The transition from the adhesive contact with contact zone to the adhesive contact without contact zone with following pull-off the bodies requires applying tensile tractions in case of strong adhesive stress.</p>