This thesis establishes a stochastic framework for tracking curves in visual clutter, using a Bayesian random-sampling algorithm. The approach is rooted in ideas from statistics, control theory and computer vision. The problem is to track outlines and features of foreground objects, modelled as curves, as they move in substantial clutter, and to do it at, or close to, video frame-rate. The algorithm, named Condensation, for Conditional density propagation, has recently been derived independently by several researchers, and is generating significant interest in the statistics and signal processing communities. This thesis contributes to the literature on Condensation-like filters by presenting some novel applications of and extensions to the basic algorithm, and contributes to the visual motion estimation literature by demonstrating high tracking performance in cluttered environments. Despite its power the Condensation algorithm has a remarkably simple form and this allows the use of non-linear motion models which combine characteristics of discrete Hidden Markov Models with the continuous Auto-Regressive Process motion models traditionally used in Kalman filters. These mixed discrete-continuous models have promising applications to the emerging field of perception of action. This thesis also implements two algorithms to smooth the output of the Condensation filter which improves the accuracy of motion estimation in a batch-mode procedure after tracking is complete.