The aim of these notes is twofold. On the one hand, we want to give an introduction to some parts of the new representation theory of finite dimensional algebras as it has been developed by the joint effort of several mathematicians through the last 15 years. We will present several of the basic methods in a unified way. We try to give a full account for those results which are available not too easily, and we will review rather carefully also the remaining results which are needed. On the other hand, we also want to exhibit the structure of the module categories of an exceptional class of algebras which we call tubular. These are very special algebras having 6, 8, 9, or 10 simple modules, but their representation theory may turn out to be of wider interest. The topics chosen in the introductory parts are those needed for an understanding of the module category of a tubular algebra, however they will be presented in greater generality, so to be useful also for other problems.

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