This work recreates the solutions to the five greatest mathematical problems of all time: The Four-Colour Map Problem, Fermat's Last Theorem, The Continuum Hypothesis, Kepler's Conjecture, and Hilbert's Tenth Problem. The author recounts these mathematical quests emphasizing the human aspect. In retelling the story of Hilbert's Tenth Problem, for instance, he sweeps from Britain to New York to Leningrad and introduces us to such luminaries as Alan Turing, before turning to the young Soviet researcher who credited his breakthrough to a 700 year-old Italian problem about rabbits. He describes how Fermat's Last Theorem tantalized generations of scientists, who tried for three centuries to answer it, and relates how the final solution was greeted with the unprecedented front-page headlines, prize money, and international celebration - before a flaw (soon resolved) turned up. Casti's account of the struggle to solve Kepler's Conjecture wittily reveals how the proof of the obvious sometimes eludes us for centuries.