Let A be an associative algebra over a commutative ring R with 1. We say that A is graded by a finite abelian group H if A =

⊕ h∈H Ah where

each Ah is an R-submodule of A and Ah1Ah2 ⊂ Ah1h2 for all h1, h2 ∈ H. Our main concern in this paper will be the determination of all possible gradings by abelian groups on integral group rings, that is, when A = ZG and H is abelian. This paper continues our research on gradings in groups rings starter in