We demonstrate that a recently developed approach for optimizing atomic structures is very effective for attacking the Thomson problem of finding the lowest-energy configuration of $N$ point charges on a unit sphere. Our approach uses a genetic algorithm, combined with a "cut and paste" scheme of mating, that efficiently explores the different low-energy structures. Not only have we reproduced the known results for $10l~Nl~132$, this approach has allowed us to extend the calculation for all $Nl~200$. This has allowed us to identify series of "magic" numbers, where the lowest-energy structures are particularly stable. Most of these structures are icosahedral, but we also find low-energy structures that deviate from icosahedral symmetry.