Abstract: "Generalised sheaf cohomology theory", by. J.F. Jardine This is an expanded version of notes for a set of lectures given at the Newton Institute during a NATO ASI Workshop entitled ``Homotopy Theory of Geometric Categories'' on September 23 and 24, 2002. The paper presents some of the basic features of the homotopy theory of simplicial presheaves and the stable homotopy theory of presheaves of spectra, and then displays their use in the course of giving an outline of proof of Thomason's descent theorem for Bott periodic K-theory, in the context of equivariant stable categories for profinite groups. Department of Mathematics University of Western Ontario London, Ontario N6A 5B7 Canada URL: http://www.math.uwo.ca/~jardine/papers/