This book is concerned mainly with the initial value problem for nonlinear hyperbolic partial differential equations. Some basic methods and techniques of studying hyperbolic equations are presented. In the book, global existence , singularities , blow up and life span of classical solutions to hyperbolic equations are discussed. Some related problems for hyperbolic geometric flow on Riemann surfaces are studied. The book can be used for students and postgraduates majoring in mathematics, teachers，scientists and engineers.