Over the last few years, the rapid advancement of 3D scanning technologies has brought huge amount of high-fidelity surface models and volumetric data sets. To effectively model, analyze and reuse these models become research thrusts in computer graphics, CAGD, vision, visualization, and virtual reality. This book covers a fundamental problem called shape mapping, i.e. computing bijective correspondence between shapes, which can facilitate many of aforementioned tasks. One direct application of mapping is non-rigid alignment for shape comparison/retrieval or deformation tracking. Another application is transforming properties, computation, or analysis of one object to another object (domain) for more efficient or robust process. Despite its importance, existing parameterization methods are not adequate to map general shapes with complicated topology and geometry. Recent research from the author of finding effective mapping methods for 2D and 3D manifolds is discussed here. It not only provides fundamental progress in understanding and computing this fundamental geometric problem, but also explores great potential of this research in broad areas of geometric and visual computing.