© 2015 IEEE. Asymptotically-optimal sampling-based motion planners, like RRT∗, perform vast amounts of collision checking, and are hence rather slow to converge in complex problems where collision checking is relatively expensive. This paper presents two novel motion planners, Lazy-PRM∗ and Lazy-RRG∗, that eliminate the majority of collision checks using a lazy strategy. They are sampling-based, any-time, and asymptotically complete algorithms that grow a network of feasible vertices connected by edges. Edges are not immediately checked for collision, but rather are checked only when a better path to the goal is found. This strategy avoids checking the vast majority of edges that have no chance of being on an optimal path. Experiments show that the new methods converge toward the optimum substantially faster than existing planners on rigid body path planning and robot manipulation problems.