When we lower the temperature of a liquid, at some point we meet a first order phase transition to the crystal. Yet, under certain conditions it is possible to keep the system in its metastable phase and to avoid crystallization. In this way the liquid enters in the supercooled phase. Supercooled liquids have a very rich phenomenology, which is still far from being completely understood. To begin with, there is the problem of how to prevent crystallization and how deeply the liquid can be supercooled before a metastability limit is hit. But by far the most interesting feature of supercooled liquids is the dynamic glass transition: when the temperature is decreased below a certain point, the relaxation time increases so much that a dramatic dynamical arrest intervenes and we are unable to equilibrate the system within reasonable experimental times. The glass transition is a phenomenon whose physical origin has stirred an enormous interest in the last hundred years. Why does it occur? Is it just a conventional reference point, or does it have a more profound physical meaning? Is it a purely dynamical event, or the manifestation of a true thermodynamic transition? What is the correlation length associated to the sharp increase of the relaxation time? Can we define a new kind of amorphous order? A shared theory of supercooled liquids and the glass transition does not yet exist and these questions are still largely open. Here, I will illustrate in the most elementary fashion the main phenomenological traits of supercooled liquids and discuss in a very partial way a few theoretical ideas on the subject.

73 pages, 20 figures