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Drinfeld's publication "Elliptic Modules" (1976) opened up the new area of study in arithmetical algebraic geometry, now called "Drinfeld modules". These objects live in positive characteristics and have analogies with elliptic curces and abelian varieties. Rank one modules are related with class field theory of function fields; rank two Drinfeld modules are analogues of elliptic curves. Their study leads to a proof of the Shimura-Taniyama-Weil conjecture and to Jacquet-Langlands theory for function fields. This book constitutes an instructional course on the subject, complemented by research papers.