34 - Poincare conjecture

In 1904, Poincaré conjectured that any closed bounded space (compact manifold) where you could pull tight all loops must be a 3-sphere.

In 2000, the Clay Mathematics Institute offered a $1,000,000 prize for proving this.

In 2003 Grigori Perelman proved the result. But, he didn't want the prize. He also refused the Fields Medal. (He's still living with his mother.)