We propose a dynamical model-based approach for tracking the shape and deformation of highly deforming objects from time-varying imagery. Previous works have assumed that the object deformation is smooth, which is realistic for the tracking problem, but most have restricted the deformation to belong to a finite-dimensional group, such as affine motions, or to finitely-parameterized models. This, however, limits the accuracy of the tracking scheme. We exploit the smoothness assumption implicit in previous work, but we lift the restriction to finite-dimensional motions/deformations. To do so, we derive analytical tools to define a dynamical model on the (infinitedimensional) space of curves. To demonstrate the application of these ideas to object tracking, we construct a simple dynamical model on shapes, which is a first-order approximation to any dynamical system. We then derive an associated nonlinear filter that estimates and predicts the shape and deformation of a object from image measurements.