My aim is to summarize the main ideas of [25] for a relatively wide audience and to communicate the structure of the proof to non-specialists. The discussion is inevitably technical at points, however, since a large amount of machinery from arithmetical algebraic geometry is required. The reader interested in a genuinely non-technical overview may prefer to begin with Mazur’s delightful introduction [19] to the Taniyama-Shimura conjecture, and to relations with Fermat’s Last Theorem and similar problems. Another excellent alternative source is the Bourbaki seminar of Oesterl´e [21]. This seminar discusses the relation between elliptic curves and Fermat’s Last Theorem from several points of view, but gives fewer details about the argument of [25] than the present summary. See also [11].