  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.
