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Many problems in pure and applied mathematics boil down to determining the shape of a surface in space or constructing surfaces with prescribed geometric properties. These problems range from classical problems in geometry, elasticity, and capillarity to problems in computer vision, medical imaging, and graphics. There has been a sustained effort to understand these questions, but many problems remain open or only partially solved. These include determining the shape of a surface from its metric and mean curvature (Bonnet's problem), determining an immersion from the projectivised Gauss map (Christoffel's problem) and its applications to the computer vision problem on recovering shape from shading, the construction of surfaces with prescribed curvature properties, constructing extremal surfaces and interfaces, and representing surface deformations.This book studies these questions by presenting a theory applying to both global and local questions and emphasizing conformal immersions rather than isometric immersions. The book offers: a unified and comprehensive presentation of the quaternionic and spinor approach to the theory of surface immersions in three and four dimensional space; new geometric invariants of surfaces in space and new open problems; a new perspective and new results on the classical geometric problems of surface and surface shape recognition and surface representation; a source of problems to motivate research and dissertations; applications in computer vision and computer graphics; and proofs of many results presented by the authors at colloquia, conferences, and congresses over the past two years.This book describes how to use quaternions and spinors to study conformal immersions of Riemann surfaces into $\Bbb R^3$. The first part develops the necessary quaternionic calculus on surfaces, its application to surface theory and the study of conformal immersions and spinor transforms. The integrability conditions for spinor transforms lead naturally to Dirac spinors and their application to conformal immersions. The second part presents a complete spinor calculus on a Riemann surface, the definition of a conformal Dirac operator, and a generalized Weierstrass representation valid for all surfaces. This theory is used to investigate first, to what extent a surface is determined by its tangent plane distribution, and second, to what extent curvature determines the shape. The book is geared toward graduate students and research mathematicians interested in differential geometry and geometric analysis and its applications, computer science, computer vision, and computer graphics.