a b s t r a c tWe introduce a generalization of the notion of local cohomology module, which we call a local cohomology module with respect to a pair of ideals (I, J), and study its various properties. Some vanishing and nonvanishing theorems are given for this generalized version of local cohomology. We also discuss its connection with ordinary local cohomology.Local cohomology theory has been an indispensable and significant tool in commutative algebra and algebraic geometry. In this paper, we introduce a generalization of the notion of local cohomology module, which we call a local cohomology module with respect to a pair of ideals (I, J), and study its various properties.To be more precise, let R be a commutative noetherian ring and let I and J be ideals of R. We are concerned with the subsetDefinition 1.5 and Corollary 1.8 (1). In general, W (I, J) is closed under specialization, but not necessarily a closed subset of Spec(R). For an R-module M, we consider the (I, J)-torsion submodule Γ I, J (M) of M which consists of all elements x of M with Supp(Rx) ⊆ W (I, J). Furthermore, for an integer i, we define the ith local cohomology functor H i I, J with respect to (I, J) to be the ith right derived functor of Γ I, J . We call H i I, J (M) the ith local cohomology module of M with respect to (I, J). See Definitions 1.1 and 1.3. Note that if J = 0 then H i I, J coincides with the ordinary local cohomology functor H i I with the support in the closed subset V (I). On the other hand, if J contains I then it is easy to see that Γ I, J is the identity functor and H i I, J = 0 for i > 0. Thus we may consider the local cohomology functor H i I, J as a family of functors with parameter J, which connects the ordinary local cohomology functor H i I with the trivial one.Our main motivation for this generalization is the following. Let (R, m) be a local ring and let I be an ideal of R. We assume that R is a complete local ring for simplicity. For a finitely generated R-module M of dimension r, Schenzel [18] introduces the notion of the canonical module K M , and he proves the existence of a monomorphism H r I (M) ∨ → K M and determines the image of this mapping, where ∨ denotes the Matlis dual. By his result, we can see that the image is actually equal to * It is obvious that if J = 0, then the (I, J)-torsion functor Γ I, J coincides with I-torsion functor Γ I . Lemma 1.2. The (I, J)-torsion functor Γ I, J is a left exact functor on the category of all R-modules. Proof. Let 0 → L f → M g → N → 0 be an exact sequence of R-modules. We must show that 0 − − → Γ I, J (L) Γ I, J (f )