In everyday life, conjectures are cheap; in mathematics, they sometimes stimulate years of vigorous theorizing. In the case of Henri Poincaré (1854-1912), the supposition was simply stated. "Consider a compact 3-dimensional manifold V without boundary," he wrote. "Is it possible that the fundamental group of V could be trivial, even though V is not homeomorphic to the 3-dimensional sphere?" At the time of his death, this pregnant thought was still in gestation. It would take almost a century for a mathematician to offer a persuasive proof of this major conundrum. Written for nonprofessional readers, The Poincaré Conjecture explains why the issue has loomed so large in scientific circles and also provides stimulating capsule biographies of the thinkers who prepared the way for its solution.