The glass problem is notoriously hard, but the recent exact solution of a microscopic model offers a novel perspective on the problem. In this seminar, I will discuss how contrasting entropic caging and isostaticity at the glass and the jamming transitions, respectively, reveals the presence of a Gardner transition. This onset of mechanical marginality then explains the presence of non-trivial critical exponents. I will also discuss how a family of finite-dimensional models reveals the clear role for caging geometry and hopping in the dynamical slowdown of colloid-like glass formers. Both advances greatly enrich the traditional mean-field description of glasses.