Dynamic surface grasping is applicable to landing of micro air vehicles (MAVs) and to grappling objects in space. In both applications, the grasper must absorb the kinetic energy of a moving object and provide secure attachment to a surface using, for example, gecko-inspired directional adhesives. Functional principles of dynamic surface grasping are presented, and two prototype grasper designs are discussed. Computer simulation and physical testing confirms the expected relationships concerning (i) the alignment of the grasper at initial contact, (ii) the absorption of energy during collision and rebound, and (iii) the force limits of synthetic directional adhesives.