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Abstract: This paper is lead by the question whether there is a nice structure theory of finitely generated modules over the Iwasawa algebra, i.e. the completed group algebra, R of a p-adic analytic group G. For G without any p-torsion element we prove that R is an Auslander regular ring. This result enables us to give a good definition for pseudo-null R-modules. Then the category of R-modules up to pseudo-isomorphisms is studied and we obtain a weak structure theorem for the p-primary part of a finitely generated R-module. A local duality theorem as well as the Auslander-Buchsbaum equality are further main issues. The arithmetic applications to the Iwasawa theory of abelian varieties are published in a forthcoming paper.