Geometric mechanics involves the use of differential geometry and symmetry techniques to study mechanical systems. In particular, it deals with global invariants of the motion, and how they can be used to describe and understand the qualitative properties of complicated dynamical systems, without necessarily explicitly solving the equations of motion. This approach parallels the development of geometric numerical methods in numerical analysis, wherein numerical algorithms for the solution of differential equations are constructed so as to exactly conserve the invariants of motion of the continuous dynamical system. This talk will provide a gentle introduction to the role of geometric methods in understanding nonlinear dynamical systems, and why it is important to develop numerical methods that have good global properties, as opposed to just good local behavior.