Numerical simulations of complex physical processes are of increasing importance in science as well as in technical applications. Usually, the mathematical treatment of such problems is founded on a-posteriori error estimation: For given data from a physical and/or technical problems one aims to obtain a numeric solution and an estimate for the error, where in usual applications in practice the exact solution remains unknown. In this book, we exemplary discuss linear parabolic differential equations with space- and time-dependent coefficients. After preparing the mathematical grounds for the later chapters, we derive an a-posteriori error estimate which is based on a work of Verfürth. Then, a space- and time-adaptive algorithm is deduced thereof, which is finally tested on several problems including the heat equation and a model concerning the spreading of pollution due to groundwater flow. The analytic considerations and model problems are accompanied with several figures and numerous explained Matlab code listings. Although focussed on numerical mathematics, readers interested in algorithms, mathematical simulation and Matlab-code implementation may find this book valuable.