  Replica symmetry breaking postulates that near optima of spin glass
Hamiltonians have an ultrametric structure. Namely, near optima can be
associated to leaves of a tree, and the Euclidean distance between them
corresponds to the distance along this tree. We survey recent progress towards
a rigorous proof of this picture in the context of mixed $p$-spin spin glass
models. We focus in particular on the following topics: $(i)$~The structure of
critical points of the Hamiltonian; $(ii)$~The realization of the ultrametric
tree as near optima of a suitable TAP free energy; $(iii)$~The construction of
efficient optimization algorithm that exploits this picture.
