This book surveys the mathematics of Galerkin finite element method as applied to parabolic equations. The approach is to first discretize in the spatial variables by Galerkin's method, using piecewise polynomial trial functions, and then to apply some time stepping method, most often of single step but also of multistep type, to determine a fully discrete solution. The concern is stability and error analysis of approximate solutions in various norms, and under various regularity assumptions on the exact solution. The book aims at giving insight in current ideas and methods of analysis rather than at pursuing each approach to its limit. It is essentially self-contained, and the use of simple model situations makes it easily accessible also for beginners in the field.