This book provides insight into the mathematics of the Galerkin finite element method as applied to parabolic equations. The approach is based on first discretizing in the spatial variables by Galerkin's method, using piecewise polynomial trial functions, and then applying some single step or multistep time-stepping methods. The concern is stability and error analysis of approximate solutions in various forms, and under various regularity assumptions on the exact solution. The book gives an excellent insight into the present ideas and methods of analysis. The second edition has been influenced by recent progress in the application of semigroup theory to stability and error analysis, particularly in maximum-norm. Two new chapters have also been added, dealing with problems in polygonal, particularly nonconvex, spatial domains, and with time discretization based on using Laplace transformation and quadrature.