Throughout this paper ‘ring’ means commutative ring with identity and modules are unital. Given rings are usually assumed to be Noetherian, but there are notable exceptions. The phrase ‘characteristic p’ always means ‘positive prime characteristic p’, and the letters q, q’, etc. denote pe, p”, etc., where e, e’ E N, the nonnegative integers. The authors have recently introduced the notion of tight closure for a submodule N of a finitely generated module M over a Noetherian ring R of characteristic p and in certain equicharacteristic zero cases, including affine algebras over fields of characteristic 0. The theory started with the study of the notion of tight closure for an ideal I c R, i.e. with the case M = R, N = I, and this is still perhaps the most important case. The notion of tight closure has yielded new proofs, and, in many instances, unexpectedly strong improvements, of the local homological conjectures, of the existence of big Cohen-Macaulay modules, of the Cohen-Macaulay property for subrings which are direct summands of regular rings (where A is a direct summand of R means that A is a direct summand of R as an A-module) in the equicharacteristic case (in fact, tight closure techniques give the first proof of this fact in complete generality in the