The geometric approach to mechanics serves as the theoretical underpinning of innovative control methodologies in geometric control theory. These techniques allow the attitude of satellites to be controlled using changes in its shape, as opposed to chemical propulsion, and are the basis for understanding the ability of a falling cat to always land on its feet, even when released in an inverted orientation. \vskip .1in \noindent Curiously, while the geometric structure of mechanical systems plays a critical role in the construction of geometric control algorithms, these algorithms have typically been implemented using numerical schemes that ignore the underlying geometry. \vskip .1in \noindent Geometric integration is the field of numerical analysis that focuses on developing geometric structure-preserving integrators, and computational geometric mechanics focuses on developing geometric integrators for dynamical systems arising from mechanics. \vskip .1in \noindent This talk will introduce some of the discrete differential geometric tools necessary to implement control algorithms in a manner that respects and preserves the geometry of the problem. These tools include discrete analogues of Lagrangian mechanics, the exterior calculus of differential forms, and connections on principal bundles. \vskip .1in \noindent This is joint work with Mathieu Desbrun (CS, Caltech), Anil Hirani (JPL), Jerrold Marsden (CDS, Caltech), Alan Weinstein (Math, Berkeley).