This will be a survey talk on subjects related to the J-homomorphism. In addition to the original geometric construction which assigns to a vector bundle its associated spherical fibration, we will touch on more algebraic constructions derived form it, in particular via Picard and unit spectra. We will examine the map from a computational point of view, studying its p-localisation and interpretation in terms of K(1)-local homotopy theory. We will almost certainly fail to do justice to any of these ideas in the time available. At the end, we will briefly extend some of these results to a higher chromatic setting.