In winter 71/72 I held a seminar on general equilibrium theory for a jOint group of students in mathematics and in econo mics at the university of Bonn , w.Germany1~ The economists , how ever , had a mathematical background well above the average * Most of the material treated in that seminar is described in these notes. The connection between smooth preferences and smooth demand func tions [ see Debreu (1972) ] and regular economies based on agents with smooth preferences are not presented here * Some pedagogical difficulties arose from the fact that elementary knowledge of algebraic topology is not assumed although it is helpful and indeed necessary to make some arguments precise * It is only a minor restriction , at present , that functional ana lysis is not used * But with the development of the theory more economic questions will be considered in their natural infinite dimensional setting * Economic knowledge is not required , but especially a reader without economic background will gain much by reading Debreu's classic Theory of Value (1959) * Although the formulation of our economic problem uses a map between Euclidean spaces only , we shall also consider ma- folds * Manifolds appear in our situation because inverse images under differentiable mappings between Euclidean spaces are very often differentiable manifolds * ( Under differentiability assump tions , for instance , the graph of the equilibrium set correspon