AbstractWe introduce first the notion of mixed multiplicities for arbitrary ideals in a local d-dimensional Noetherian ring (A,m) which, in some sense, generalizes the concept of mixed multiplicities for m-primary ideals. We also generalize Teissier's product formula for a set of arbitrary ideals and extend the notion of the Buchsbaum-Rim multiplicity (BR-multiplicity) of a submodule of a free module to the case where the submodule no longer has finite colength. For a submodule M of A p , we introduce a sequence of multiplicities e k BR (M), k = 0, . . . , d + p − 1 which in the case of an ideal (p = 1) coincides with the multiplicity sequence c0(I,A), . . . , c d (I,A) defined for an arbitrary ideal I of A by Achilles and Manaresi. In the case where M has finite colength in Ap and is totally decomposable, we prove that our BR-multiplicity sequence essentially falls into the standard BR-multiplicity of M.