This book contains a systematic and comprehensive exposition of Lobachevskian geometry and the theory of discrete groups of motions in Euclidean space and Lobachevsky space.It is divided into two closely related parts： the first treats the geometry of spaces of constant curvature and the second discrete groups of motions of these.The authors give a very clear account of their subject describing it from the viewpoints of elementary geometry， Riemannian geometry and group theory.The result is a book which has no rival in the literature.Part I contains the classification of motions in spaces of constant curvature and non-traditional topics like the theory of acute-angled polyhedra and methods for computing volumes of non-Euclidean polyhedra.Part II includes the theory of cristallographic， Fuchsian， and Kleinian groups and an exposition of Thurstonis theory of deformations.The greater part of the book is accessible to first-year students in mathematics.At the same time the book includes very recent results which will be of interest to researchers in this field.
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