Solving robotic tasks involves building planners able to deal with uncertainty. This talk shows how combinatorial topology informs that planning process by characterizing the capabilities of robotic systems topologically. Planning problems, particularly for robotic manipulation tasks, may often be cast as discrete graph search problems. Motions in the graph are generally nondeterministic or stochastic as a result of control and sensing errors in the underlying physical system. The global capabilities of such a system may be modeled by the homotopy type of a "strategy complex". The simplices of this complex may be viewed as the eventually-convergent control laws (aka plans or strategies) possible on the graph. One result is a controllability theorem: The robot can move anywhere in the graph despite control uncertainty precisely when the graph's strategy complex is homotopic to a sphere of a certain dimension. The strategy complex resides in a space of motions, but one can compress it back onto the graph, without losing homotopy type. Viewed on the graph, nonsimplices of the simplicial complex have intuitive meaning: they represent potentially inescapable regions of the graph. One can further understand the capabilities of a system via open covers whose open sets are intersections of Euclidean halfspaces determined by the possible motions and their execution times. The nerve of such a cover has the same homotopy type as a time- bounded variant of the strategy complex.