We study the structure of the categories of K(n)-local and E(n)-local spectra, using the axiomatic framework developed in earlier work of the authors with John Palmieri. We classify localising and colocalising subcat-egories, and give characterisations of small, dualisable, and K(n)-nilpotent spectra. We give a number of useful extensions to the theory of vn self maps of finite spectra, and to the theory of Landweber exactness. We show that certain rings of cohomology operations are left Noetherian, and deduce some powerful finiteness results. We study the Picard group of invertible K(n)-local spectra, and the problem of grading homotopy groups over it. We prove (as announced by Hopkins and Gross) that the Brown-Comenetz dual of MnS lies in the Picard group. We give a detailed analysis of some examples when n = 1 or 2, and a list of open problems.