This paper is motivated by the question whether there is a nice structure theory of finitely generated modules over the Iwasawa algebra, i.e. the completed group algebra, 6 of a p-adic analytic group G. For G without any p-torsion element we prove that 6 is an Auslander regular ring. This result enables us to give a good definition of the notion of a pseudo-null 6-module. This is classical when G=Êkp for some integer kS1, but was previously unknown in the non-commutative case. Then the category of 6-modules up to pseudo-isomorphisms is studied and we obtain a weak structure theorem for the Êp-torsion part of a finitely generated 6-module. We also prove a local duality theorem and a version of Auslander-Buchsbaum equality. The arithmetic applications to the Iwasawa theory of abelian varieties are published elsewhere.