This outstanding textbook by a distinguished mathematical scholar introduces the differential geometry of curves and surfaces in three-dimensional Euclidean space. The subject is presented in its simplest, most essential form, but with many explanatory details, figures and examples, and in a manner that conveys the geometric significance and theoretical and practical importance of the different concepts, methods and results involved. The first chapters of the book focus on the basic concepts and facts of analytic geometry, the theory of space curves, and the foundations of the theory of surfaces, including problems closely related to the first and second fundamental forms. The treatment of the theory of surfaces makes full use of the tensor calculus. The later chapters address geodesics, mappings of surfaces, special surfaces, and the absolute differential calculus and the displacement of Levi-Cività. Problems at the end of each section (with solutions at the end of the book) will help students meaningfully review the material presented, and familiarize themselves with the manner of reasoning in differential geometry.