Posed over a century ago as a model of electronic structure within atoms, the so-called Thomson problem of determining the minimum-energy configuration of identical point charges in the unit sphere remains a well-known open problem in statistical physics. In the related Tammes problem, we seek an arrangement of points on a sphere which maximizes the smallest distance between any pair of points. Global optima are only known for very small instances of the Thomson and Tammes problems. In this paper, we present a branch-and-bound algorithm for these problems which exploits their geometry and dynamically eliminates symmetric subproblems. Our algorithm shows the global optimality within numerical tolerance of the putative solution to the 7-electron instance of the Thomson problem for the first time. For the Tammes problem, we recover the established results for instances with up to 13 points.