. We study phantom maps and homology theories in a stable homotopy category S via a certain Abelian category A. We express the group P(X; Y ) of phantom maps X ! Y as an Ext group in A, and give conditions on X or Y which guarantee that it vanishes. We also determine P(X; HB). We show that any composite of two phantom maps is zero, and use this to reduce Margolis's axiomatisation conjecture to an extension problem. We show that a certain functor S ! A is the universal example of a homology theory with values in an AB 5 category and compare this with some results of Freyd. Contents 1. Introduction 1 2. Axiomatic stable homotopy theory 4 3. Homology theories 6 4. Phantom maps 9 5. Margolis's axiomatisation conjecture 15 6. Phantom cohomology 16 7. Universal homology theories 21 References 25 1. Introduction In this paper we collect together a number of results about the homotopy category of spectra. A central theme is the problem of reconstructing this category from the category of n...