Keywords: Shape evolution; Shape spaces; Active contours; Riemannian metrics on diffeomorphisms Abstract: We present a geometric flow approach to the segmentation of two- three- dimensional shapes by minimizing a cost function similar to the ones used with geometric active contours or to the Chan-Vese approach. Our goal, well-adapted to many shape segmentation problems, including those arising from medical images, is to ensure that the evolving contour or surface remains smooth and diffeomorphic to its initial position over time. This is formulated as a variational problem in a group of diffeomorphisms equipped with a right-invariant Riemannian metric. A resulting gradient descent algorithm is used, with an interesting variant that constrains the velocity in shape space to belong in a finite dimensional space determined by time-dependent control points. Experimental results with 2D and 3D shapes will be presented.