In optical design, the matrix method is used to find a solution to a given optical task, which can then be refined by optical-design software or analytical methods of aberration balancing. In some cases, the method can be helpful to demonstrate that there is no solution possible under the given boundary conditions. Quite often it is of practical importance and theoretical interest to get an overview on the "solution space" of a problem. The paraxial approach might then serve as a guideline during optimization in a similar way as a map does in an unknown landscape. The course familiarizes attendees with the application of the method of transfer matrices and related techniques to a variety of optical engineering problems. After an introduction to the method, it describes applications to imaging optics as well as to illumination systems. The course concentrates on devices of practical importance as zoom systems, interferometric devices, and laser resonators. Emphasis is also on providing a toolbox for first-order tolerancing and sensitivity analysis. The course comprises the analysis of anamorphic optical devices, because of their growing market penetration.