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This book, which was originally published in 1985 and has been translated and revised by the author from notes of a course, is an introduction to certain central ideas in group theory and geometry. Professor Lyndon emphasises and exploits the well-known connections between the two subjects and, whilst keeping the presentation at a level that assumes only a basic background in mathematics, leads the reader to the frontiers of current research at the time of publication. The treatment is concrete and combinatorial with a minimal use of analytic geometry. In the interest of the reader's intuition, most of the geometry considered is two-dimensional and there is an emphasis on examples, both in the text and in the problems at the end of each chapter.