Over the past few years, shape memory polymer (SMP) has been extensively studied in terms of its remarkable reversible dry adhesive properties and related smart adhesive applications. However, its exceptional properties have not been exploited for further opportunities such as pick-and-place applications, which would otherwise advance the robotic manipulation. This work explores the use of an SMP to design an adhesive gripper that picks and places a target solid object employing the reversible dry adhesion of an SMP. Compared with other single surface contact grippers including vacuum, electromagnetic, electroadhesion, and gecko grippers, the SMP adhesive gripper interacts with not only flat and smooth dry surfaces but also moderately rough and even wet surfaces for pick-and-place with high adhesion strength (> 2 atmospheres). In this work, associated physical mechanisms, SMP adhesive mechanics, and thermal conditions are studied. In particular, the numerical and experimental study elucidates that the optimal compositional and topological SMP design may substantially enhance its adhesion strength and reversibility, which leads to a strong grip force simultaneously with a minimized releasing force. Finally, the versatility and utility of the SMP adhesive gripper are highlighted through diverse pick-and-place demonstrations.