We briefly review the nonlinear dynamics of diverse physical systems which can be described in terms of moving curves and surfaces. The interesting connections that exist between the underlying differential geometry of these systems and the corresponding nonlinear partial differential equations are highlighted by considering classic examples such as the motion of a vortex filament in a fluid and the dynamics of a spin chain. The association of the dynamics of a non-stretching curve with a hierarchy of completely integrable soliton-supporting equations is discussed. The application of the surface embeddability approach is shown to be useful in obtaining such connections as well as exact solutions of some nonlinear systems such as the Belavin-Polyakov equation and the inhomogeneous Heisenberg chain.