We add to a manipulator’s capabilities a new primitive motion which we term a push-grasp. While significant progress has been made in robotic grasping of objects and geometric path planning for manipulation, such work treats the world and the object being grasped as immovable, often declaring failure when simple motions of the object could produce success. We analyze the mechanics of push-grasping and present a quasi-static tool that can be used both for analysis and simulation. We utilize this analysis to derive a fast, feasible motion planning algorithm that produces stable push-grasp plans for dexterous hands in the presence of object pose uncertainty and high clutter. We demonstrate our algorithm extensively in simulation and on HERB, a personal robotics platform developed at Intel Labs Pittsburgh.