We investigate the premise that robust grasping performance is enabled by exploiting constraints present in the environment. These constraints, leveraged through motion in contact, counteract uncertainty in state variables relevant to grasp success. Given this premise, grasping becomes a process of successive exploitation of environmental constraints, until a successful grasp has been established. We present support for this view found through the analysis of human grasp behavior and by showing robust robotic grasping based on constraint-exploiting grasp strategies. Furthermore, we show that it is possible to design robotic hands with inherent capabilities for the exploitation of environmental constraints.