This chapter describes a set of techniques for finding correspondences and computing registrations between 3D objects, which deform in a nonrigid way. It provides a general formulation and lay down the mathematical framework. The chapter describes some important mathematical concepts and tools on which the methods presented in the chapter are based upon. It focuses on methods which assume that the type of deformations 3D objects undergo are limited to bending, referred to as isometric deformations. The chapter turns our attention to the general problem where deformations include bending and stretching. It discusses different deformation models and how the optimal deformation path (i.e. registration) can be computed under fixed rotation and parameterization. The chapter describes one representation called square-root normal fields (SRNFs), which brings the complex elastic deformation models into a more simple and computationally tractable representation. It discusses how elastic correspondence and registration are performed under the SRNF representation.