This text is designed for a one-semester undergraduate course in projective geometry. In incorporates a synthetic approach starting with axioms from which the general theory is deduced, together with an analytic approach using the real projective plane as a model. The first is refined as the second is generalized until the two coincide via the introduction of coordinates in an abstract projective plane. Special attention is paid to the role of Desargues' and Pappus' axioms in the theory. At the end of the book is a list of problems that can be used as exercises while reading. The emphasis on the various groups of transformations that arise in projective geometry introduces the reader to group theory in a practical context. While the book does not assume any previous knowledge of abstract algebra, some familiarity with group theory would be useful. First published in 1967 and long out of print, this book is now reissued with a new preface, an appendix on the simple group of order 168, which appears as the group of automorphisms of a projective plane of seven points, and a list of errata.