The dynamics of liquids, regarded as strongly-interacting classical particle systems, remains a field where theoretical descriptions are limited. So far, there is no microscopic theory starting from first principles and using controlled approximations. At the thermodynamic level, static equilibrium properties are well understood in simple liquids only far from glassy regimes. In this seminar, I will introduce the long-standing problems of the dynamics of classical liquids and the glass transition. I will review the insights obtained from the study of mean-field spin glasses, giving birth to a generic scenario to glassy systems, the random first-order transition (RFOT) theory. On this basis, I will present the recent derivation of the dynamics of liquids and structural glasses using the limit of large spatial dimension, which provides a well-defined mean-field approximation with a clear small parameter. In parallel, one recovers their thermodynamics through an analogy between dynamics and statics. This gives a unifying and consistent view of the phase diagram of these systems. We showed that this mean-field solution to the structural glass problem is an example of the RFOT scenario, as conjectured thirty years ago. These results allow to show that an approximate scale invariance of the system, relevant to finite-dimensional experiments and simulations, becomes exact in this limit.