This is a book about the classification theory and geometry of higher dimensional varieties: existence and geometry of rational curves via characteristic p-methods, manifolds with negative Kodaira dimension, vanishing theorums, theory of extremal rays (Mori theory), and minimal models. It gives an introduction to this difficult and not readily accessible subject, and focuses on the spread of ideas rather than a deliberate inclusion of all proofs. The methods presented vary from complex analysis to complex algebraic geometry and algebraic geometry over fields of positive characteristics. It should be suitable for students in algebraic geometry and complex analysis as well as researchers in these fields and experts from other fields who require an overview of the theory.