  Despite decades of work, gaining a first-principle understanding of amorphous
materials remains an extremely challenging problem. However, recent theoretical
breakthroughs have led to the formulation of an exact solution in the
mean-field limit of infinite spatial dimension, and numerical simulations have
remarkably confirmed the dimensional robustness of some of the predictions.
This review describes these latest advances. More specifically, we consider the
dynamical and thermodynamic descriptions of hard spheres around the dynamical,
Gardner and jamming transitions. Comparing mean-field predictions with the
finite-dimensional simulations, we identify robust aspects of the description
and uncover its more sensitive features. We conclude with a brief overview of
ongoing research.
