I will survey recent progress on computing approximate ground states of mean-field spin glass Hamiltonians, which are certain random functions in high dimension. While the asymptotic ground state energy OPT is given by the famous Parisi formula, the landscape of H_N features many bad local optima impeding efficient optimization. In the positive direction, I will explain algorithms based on approximate message passing which achieve asymptotic value ALG given by an extended Parisi formula. The case ALG=OPT has a natural “no overlap gap” interpretation. In the negative direction, I will explain why no algorithm with suitably Lipschitz dependence on H_N can surpass ALG - this includes most standard optimization algorithms on dimension-free time scales. Based on joint works with Ahmed El Alaoui, Brice Huang, and Andrea Montanari.