This book provides a unique and highly accessible approach to singularity theory from the perspective of differential geometry of curves and surfaces. It is written by three leading experts on the interplay between two important fields ? singularity theory and differential geometry. The book introduces singularities and their recognition theorems, and describes their applications to geometry and topology, restricting the object of attention to singularities of plane curves and surfaces in the Euclidean 3-space. In particular, by presenting the singular curvature, which is originally introduced by the authors, the Gauss?Bonnet theorem for surfaces is generalized to those with singularities. The Gauss?Bonnet theorem is intrinsic in nature, that is, it is a theorem not only for surfaces but also for 2-dimensional Riemannian manifolds. The book also illustrates the notion of Riemannian manifolds with singularities. These topics, as well as elementary descriptions of proofs of the recognition theorems, cannot be found in other books. Explicit examples and models are provided in abundance, along with insightful explanations of the underlying theory as well. Numerous figures and exercise problems are given, becoming strong aids in developing an understanding of the material. Readers will gain from this text a unique introduction to the singularities of curves and surfaces from the viewpoint of differential geometry, and it will be a useful for students and researchers interested in this subject. Key Features ? A geometric approach is taken to explain about singularities, and providing the readers with a good intuition for the behavior of singularities ? The flow of the text is carefully explained so that readers can efficiently learn from it in ways that match their individual purposes ? The book covers a spectrum of material from the most basic facts to advanced research topics, in a way that naturally brings the reader through the different levels ? The authors have written this book always with the needs of the reader in mind. Topics are introduced in stages, spanning the spectrum from basic tools to present-day research questions. The first stages provide the reader with ample amounts of intuition, before the later stages explain the important technical tools of the field ? If the reader is interested in understanding the typical singularities found on curves and surfaces in Euclidean 3-space (cusps, cuspidal edges, swallowtails, cross caps, etc), this book is an ideal source. If, rather, the reader wishes to go deeper into the field and understand singularities on submanifolds in more general ambient spaces, or understand how the Gauss?Bonnet theorem extends beautifully to cases of surfaces with singularities, this book is again a fine source ? The reader has various options for how he/she could learn and make use of the material in this book, depending on his/her goals. As a result, this book will be of interest to both the beginner and professional mathematicians alike

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