This book is an introduction to twistor theory and modern geometrical approaches to space-time structure at the graduate or advanced undergraduate level. The choice of material presented has evolved from graduate courses given in London and Oxford and the authors have aimed to retain the informal tone of those lectures. Topics covered include spinor algebra andcalculus; compactified Minkowski space; the geometry of null congruences; the geometry of twistor space; an informal account of sheaf cohomology sufficient to describe the twistor solution for the zero rest-mass equations; the active twistor constructions which solve the self-dual Yang-Mills and Einstein equations; and Penrose's quasi- local-mass construction. Exercises are included in the text and after most chapters. The book will provide graduate students with an introduction to the literature of twistor theory, presupposing some knowledge of special relativity and differential geometry. It would also be of use for a short course on space-time structure independent of twistor theory.