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We introduce the notion of a diagram category and discuss its application tothe invariant theory of classical groups and super groups, with someindications concerning extensions to quantum groups and quantum super groups.Tensor functors from various diagram categories to categories ofrepresentnations are introduced and their properties investigated, leading tofirst and second fundamental theorems of invariant theory for classical supergroups, which include the classical groups as special cases. Application ofdiagrammatic methods enables the constructionof a presentation for endomorphismalgebras for te orthogonal and symplectic groups, leading to the solutionofproblems raised by the work of Brauer and Weyl.

Diagram categories and invariant theory for classical groups and supergroups