Abstract. This is a story that involves many strands and many players, and like all such stories, it can be told in many ways. I will start by picking up a strand involving superfast wavelet transforms. As the story unfolds, other strands and players will be brought in, to discuss superfast as well as other effective ways in which to compute sparse transforms from what might, at first sight, seem incomplete data. It is a story that features many young researchers, and in which the speaker herself plays an incidental role only.