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.