This book surveys the mathematics of 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 method. The concern is stability and error analysis of approximate solutions in various norms, and under various regularity assumptions on the exact solution. The book gives an excellent insight in the present ideas and methods of analysis rather than pursuing each approach to its limit. It is essentially self-contained, and simple model situations make it easily accessible even for beginners in the field. Its basis is the author's LNM volume 1054 of 1984, which has been substantially amended.