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A shape memory polymer (SMP) has been intensively researched in terms of its exceptional reversible dry adhesive characteristics and related smart adhesive applications over the last decade. However, its unique adhesive properties have rarely been taken into account for other potential applications, such as robotic pick-and-place, which might otherwise improve robotic manipulation and contribute to the related fields. 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. The numerical and experimental results reveal that an ideal compositional and topological SMP adhesive design can significantly improve its adhesion strength and reversibility, leading to a strong grip force and a minimal release force. Next, a radially averaged power spectrum density (RAPSD) analysis proves that active heating and cooling with a thermoelectric Peltier module (TEC) substantially enhances the conformal adhesive contact of an SMP. Based on these findings, an adhesive gripper is designed, fabricated, and tested. Remarkably, the SMP adhesive gripper interacts not only with flat and smooth dry surfaces, but also moderately rough and even wet surfaces for pick-and-place, showing high adhesion strength (>2 standard atmospheres) which is comparable to or exceeds those of other single-surface contact grippers, such as vacuum, electromagnetic, electroadhesion, and gecko grippers. Lastly, the versatility and utility of the SMP adhesive gripper are highlighted through diverse pick-and-place demonstrations. Associated studies on physical mechanisms, SMP adhesive mechanics, and thermal conditions are also presented.