This book offers the first systematic presentation of the theory of the mixed problem for hyperbolic differential equations with variable coefficients. This class includes hyperbolic and parabolic equations as well as nonclassic type of operator - the $q$-hyperbolic equation - which was introduced by the authors. As part of the exposition, the authors consider the Cauchy problem for this class of equations. This book would be suitable as a graduate textbook for courses in partial differential equations.