Abstract - The glass transition is a major open question in condensed matter physics as it challenges some basic notions of materials science. Cooling a liquid fast enough below its melting temperature, the viscosity dramatically increases leading to a solid-like macroscopic behavior, whereas the microscopic liquid structure remains unchanged. The diverse theoretical approaches proposed until now only capture some partial aspects of the glassy behavior. In particular, the Mode Coupling Theory (MCT), starting from first-principles, makes a number of precise dynamical predictions. However it mistakenly predicts a divergence of the relaxation time at a relatively high temperature, where a crossover to a faster increase upon cooling is in fact observed. After a brief review of the main open questions on the glassy behavior of supercooled liquids I will focus on the crossover problem. Recent theoretical advances based on the analogy with spin-glasses offer a solution to the classic problem of the avoided MCT singularity. The slow dynamics in the crossover region is mapped into a simpler model, called Stochastic Beta Relaxation, that allows to discuss many important features of current experimental and numerical research, including dynamical susceptibilities, Stokes-Einstein violations, dynamical heterogeneities and dynamical correlation lengths.
