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