In 1890 Stickelberger published his eponymous theorem in Math. Annalen giving an explicit annihilator for the `minus' (or imaginary) part of the class group of a cyclotomic field as a Galois module. However, Stickelberger's wonderful theorem raises more questions than it answers. And strangely, many obvious ones have only begun to receive serious attention - let alone answers - in the late 20th and early 21st centuries. For instance: Is there a similar result for an arbitrary (abelian) extension of number fields? Is the `Stickelberger ideal' the full annihilator of minus part the class group? What about the `plus' (or real) part? The first question leads to Brumer's Conjecture. The answer to the second question is certainly negative, for several different reasons which we shall try to disentangle. This leads to comparisons with the Fitting ideal of the class group and of its Pontrjagin dual, and so to very recent work by Greither, the speaker and others, which we shall survey. If time allows we should like to report on some recent approaches to the third question.