Abstract : One of the aims of Computer Vision in the past thirty years has been to recognize shapes with numerical algorithms. In this chapter, we describe five curve smoothing algorithms, of growing sophistication and invariance. We give a detailed implementation and link these algorithms to the curve evolution PDE's they implement. We let the five algorithms undergo a practical invariance testing. The tested invariance requirements face no less than five classes of perturbations, namely noise, geometric or affine distortions, contrast changes, occlusion and figure/background reversal. Quite contrary to the main stream idea that curve evolution schemes should be implemented by level set methods, we describe very fast and accurate direct curve evolution implementations. We give precise bibliographical links to the mathematical and image analysis literature justifying these curve evolution algorithms and their relationship to mathematical morphology, scale space theory, classical filtering. We finally give some hints on the role of shape smoothing in actual shape recognition systems.