This book is written for theoretical and mathematical physicists and mat- maticians interested in recent developments in complex general relativity and their application to classical and quantum gravity. Calculations are presented by paying attention to those details rmally omitted in research papers, for pedagogical r- sons. Familiarity with fibre-bundle theory is certainly helpful, but in many cases I only rely on two-spir calculus and conformally invariant concepts in gravitational physics. The key concepts the book is devoted to are complex manifolds, spir techniques, conformal gravity, ?-planes, ?-surfaces, Penrose transform, complex 3 1 - - space-time models with n-vanishing torsion, spin- fields and spin- potentials. 2 2 Problems have been inserted at the end, to help the reader to check his und- standing of these topics. Thus, I can find at least four reasons for writing yet ather book on spir and twistor methods in general relativity: (i) to write a textbook useful to - ginning graduate students and research workers, where two-component spir c- culus is the unifying mathematical language.