This book consists of a series of self-contained essays in non-Euclidean geometry in a broad sense, including the classical geometries of constant curvature (spherical and hyperbolic), de Sitter, anti-de Sitter, co-Euclidean, co-Minkowski, Hermitian geometries, and some axiomatically defined geometries. Some of these essays deal with very classical questions and others address problems that are at the heart of present-day research, but all of them are concerned with fundamental topics.

All the essays are self-contained, and most of them can be understood by the general educated mathematician. They should be useful to researchers and to students of non-Euclidean geometry and are intended to be references for the various topics they present.