Humanoid robots walking across intermittent terrain, robotic arms grasping multifaceted objects, or UAVs darting left or right around a tree ... many of the dynamics and control problems we face today have both rich nonlinear dynamics and an inherently combinatorial structure. In this talk, Tedrake will review some recent work on planning and control methods which address these two challenges simultaneously. He will present our explorations with mixed-integer convex-, semidefinite-programming-relaxations, and satisfiability-modulo-theory(SMT)-based methods applied to hard problems in legged locomotion over rough terrain, grasp optimization, and UAVs flying through highly cluttered environments.