These notes were the subject of lectures given at Kent State University during the 1973-74 academic year. At that time, it was already clear that the geometry of Banach spaces (in the form of convexity and smoothness type considerations) would play a central role in the theory of Radon-Nikodym differentiation for vector-valued measures. This was the object of the course: to acquaint my students (and, to a large extent, myself) with the geometry of Banach spaces. Naturally, the logical finish to the courses was a discussion of the Radon-Nikodym theorem viewed from a purely geometric perspective.