The authors give the foundation of an eigenfunction approach (separation of variables) to the boundary value and initial boundary value problems for linear partial differential equations. The method is applied to three basic equations in the plane with non-trivial right-hand sides. In contrast to basic courses on partial differential equations where the method is usually explained very superficially, in this book it is done in full detail. Possibilities and advantages of the method are also considered in detail in this very carefully written text. Solved examples as well as many exercises of various degrees of difficulty accompany each chapter. In my opinion, the book is very inspiring and it offers a lot of themes for theses at the Bachelors and Masters level.