The mathematics of shape has a long history in the fields of differential geometry and topology. But does this theory of shape address the central problem of vision: finding the best data structure plus algorithm for storing a shape and later recognizing the same and similar shapes. Several criteria may be used to evaluate this: does the data structure capture our intuitive idea of 'similarity'? does it allow reconstruction of typical shapes to compare with new input? One direction in which mathematics and vision have converged is toward multiscale analyses of visual signals and shapes. In other respects, however, the recognition process in animals shows features that still defy mathematical modeling.