This volume introduces a unified, self-contained study of linear discrete parabolic problems through reducing the starting discrete problem to the Cauchy problem for an evolution equation in discrete time. Accessible to beginning graduate students, the book contains a general stability theory of discrete evolution equations in Banach space and gives applications of this theory to the analysis of various classes of modern discretization methods, among others, Runge-Kutta and linear multistep methods as well as operator splitting methods. Among its key features, it: presents a unified approach to examining discretization methods for parabolic equations; highlights a stability theory of discrete evolution equations (discrete semigroups) in Banach space; deals with both autonomous and non-autonomous equations as well as with equations with memory; offers a series of numerous well-posedness and convergence results for various discretization methods as applied to abstract parabolic equations; among others, Runge-Kutta and linear multistep methods as well as certain operator splitting methods; and provides comments of results and historical remarks after each chapter.