- Abstract Residue complexes were introduced by Grothendieck in algebraic geometry. These are canonical complexes of injective modules that enjoy remarkable functorial properties (traces). In this paper we study residue complexes over noncommutative rings. These objects have a more intricate structure than in the commutative case, since they are complexes of bimodules. We develop methods to prove uniqueness, existence and functoriality of residue complexes. For a polynomial identity algebra over a field (admitting a Noetherian connected filtration) we prove existence of the residue complex and describe its structure in detail.