Lie groups with left- or right-invariant metrics are a good source of examples in Riemannian geometry: simple enough to work things out explicitly, complex enough to exhibit lots of interesting behavior. Geodesics on such groups arise in applications such as the motion of a rigid body (a geodesic motion on the group SO(2), with left-invariant metric determined by the moments of inertia). Since 1966, when Arnold observed that the Euler equations for a rigid body were structurally similar to the Euler equations for an ideal fluid, the notion of viewing equations of continuum mechanics as geodesic equations on diffeomorphism groups has become important. I will discuss some of the more important equations that arise from this method (ideal hydrodynamics, Korteweg-de Vries, Camassa-Holm) and how this technique simplifies existence and uniqueness results. In addition I will discuss the global properties of the resulting exponential map. Remembering some of my previous talk will be helpful.