Introduction to Einsteins theory of relativity : from Newtons attractive gravity to the repulsive gravity of vacuum energy Newton’s Law of Universal Gravitation The Special Theory of Relativity Vectors Tensors and Forms Accelerated Reference Frames Covariant Differentiation Curvature Einstein’s Field Equations The Schwarzschild Spacetime The linear field approximation and gravitational waves Black Holes Sources of gravitational fields Cosmology. The revised and updated 2nd edition of this established textbook provides a self contained introduction to the general theory of relativity describing not only the physical principles and applications of the theory but also the mathematics needed in particular the calculus of differential forms. Updated throughout the book contains more detailed explanations and extended discussions of several conceptual points and strengthened mathematical deductions where required. It includes examples of work conducted in the ten years since the first edition of the book was published for example the pedagogically helpful concept of a river of space and a more detailed discussion of how far the principle of relativity is contained in the general theory of relativity. Also presented is a discussion of the concept of the gravitational field in Einsteins theory and some new material concerning the twin paradox in the theory of relativity. Finally the book contains a new section about gravitational waves exploring the dramatic progress in this field following the LIGO observations. Based on a long established masters course the book serves advanced undergraduate and graduate level students and also provides a useful reference for researchers.