This thesis investigates the problem of dexterous manipulation; how can robots affect the world around them by means of their end-effectors? Dexterous manipulation is fundamental to robots operating intelligently and independently in their environments and it is a special motion planning problem. Since the general motion planning problem with uncertainty is NEXP-hard, effort must be directed to defining classes of tasks that are tractable. We consider the reorientation problem: for a given robot hand, an arbitrary object, and a desired orientation with respect to the hand, find an algorithm to synthesize a robust plan for the fingers that accomplishes the desired reorientation. A reorientation algorithm devised for a robot hand should satisfy several properties. First, it must be able to accomplish arbitrarily large rotations. Second, since it must be implemented on a real device, it should involve simple finger motions that can be computed fast. Third, since the application domain is characterized by uncertainties that manifest themselves as imprecisions in calculations and inaccuracies in control, it must exhibit good stability properties. We propose algorithms for the reorientation problem that satisfy these properties. The basic idea is to use some of the robot fingers to constrain the motion of the object and others to generate motion. This results in the idea of finger tracking as a high-level primitive for manipulation. We also propose an algebraic framework for manipulation that is theoretically well-founded and in which it is possible to use systematically and effectively the differential equations describing the interaction between objects in contact. Finally, we describe a simulator for the reorientation of polyhedra by finger tracking.