To most of the outside world, mathematics appears to be a closed book: everything known, no loose ends, no mysteries. From within the subject, it is hard to understand how such an impression can have arisen, for in almost every direction today’s mathematician sees unsolved problems and wonderful new ideas. There seems to be little danger that mathematics will either grind to a halt in the face of insuperable obstacles, or become defunct when the Final Theorem is proved and nothing remains to be discovered.