Celebrated mathematician Stewart explains why mathematical problems exist, what drives mathematicians to solve them, and why their efforts matter in the context of science as a whole. Great problems—Prime territory : Goldbach Conjecture—The puzzle of pi : squaring the circle—Mapmaking mysteries : Four Color theorem—Sphereful symmetry : Kepler Conjecture—New solutions for old : Mordell Conjecture—Inadequate margins : Fermat's Last Theorem—Orbital chaos : Three-body problem—Patterns in prime : Riemann Hypothesis—What shape is a sphere? : Poincaré Conjecture—They can't all be easy : P/NP problem—Fluid thinking : Navier-Stokes Equation—Quantum conundrum : Mass Gap Hypothesis—Diophantine dreams : Birch-Swinnerton-Dyer Conjecture—Complex cycles : Hodge Conjecture—Where next? — Twelve for the future.