This book is concerned with Cauchy problem for quasilinear hyperbolic systems. By introducing the concepts weak linear degeneracy and matching condition, we give a systematic presentation on the global existence, the large time behaviour and the blow-up phenomenon, particularly, the life span of $$ solutions to the Cauchy problem with small and decaying initial data. Some successful applications of our general theory are given to the quasilinear canonical system related to the Monge-Ampére equation, the system of nonlinear three-wave interaction in plasma physics, the nonlinear wave equation with higher order dissipation, the system of one-dimensional gas dynamics with nonlinear dissipation, the system of motion of an elastic string, the system of plane elastic waves for hyperelastic materials and so on.