In this paper, we propose a new framework to perform nonrigid surface registration. It is based on various extensions of an iterative algorithm recently presented by several researchers (Besl and McKay, 1992; Champleboux et al., 1992; Chen and Medioni, 1992; Menq and Lai, 1992; Zhang, 1994) to rigidly register surfaces represented by a set of 3D points, when a prior estimate of the displacement is available. Our framework consists of three stages: - First, we search for the best rigid displacement to superpose the two surfaces. We show how to efficiently use curvatures to superpose principal frames at possible corresponding points in order to find a prior rough estimate of the displacement and initialize the iterative algorithm. - Second, we search for the best affine transformation. We introduce differential information in points coordinates: this allows us to match locally similar points. Then, we show how principal frames and curvatures are transformed by an affine transformation. Finally, we introduce this differential information in a global criterion minimized by extended Kalman filtering in order to ensure the convergence of the algorithm. - Third, we locally deform the surface. Instead of computing a global affine transformation, we attach to each point a local affine transformation varying smoothly along the surface. We call this deformation a locally affine deformation. All these stages are illustrated with experiments on various real biomedical surfaces (teeth, faces, skulls, brains and hearts), which demonstrate the validity of the approach.