This is a considerably expanded version of the first 75% of the authors Differential Geometry and Symmetric Spaces, Academic Press 1962. For the graduate and advanced undergraduate students as well as mathematicians and mathematical physicists, the new book is both a self-contained text and a reference source on each of the three topics in the title. The book provides an elementary introduction to differential geometry and to Lie groups and Lie algebras. It then focuses on the detailed structure theory of semisimple Lie groups and the geometry of symmetric spaces, ending with a complete classification. To facilitate self-study and to indicate further developments each chapter ends with "Exercises and Further Results". Solutions and references are given at the end of the book.