Abstract: We will start with an introduction to the mod 2 Steenrod algebra, and some structure theory for modules over its subalgebras due to Adams and Margolis. In particular, there is a set of homology functors which detect stable isomorphism, and there are subcategories of modules that are `local' with respect to them.

Next, we specialize to the subalgebras relevant to real and complex K-theory, called E(1) and A(1), where we can give quite precise descriptions of the local modules. The Picard groups of these subcategories are sufficient to detect the Picard group of the whole category and contain modules of geometric interest.

General results obtained along the way allow us to begin to attack the analogous questions for E(2) and A(2)-modules.

Applications include better descriptions of polynomial algebras as modules over the Steenrod algebra, and of the values of certain generalized cohomology theories on the classifying spaces of elementary abelian groups.