  These lecture notes can be read in two ways. The first two Sections contain a
review of the phenomenology of several physical systems with slow
nonequilibrium dynamics. In the Conclusions we summarize the scenario derived
from the solution to some solvable models (p-spin and the like) that are
intimately connected to the mode coupling approach (and similar ones) to
super-cooled liquids. At the end we list a number of open problems of great
relevance in this context. These Sections can be read independently of the body
of the paper where we present some of the basic analytic techniques used to
study the out of equilibrium dynamics of classical and quantum models with and
without disorder. The technical part starts wIth a brief discussion of the role
played by the environment and quenched disorder in the dynamics of classical
and quantum systems. Later on we expand on the dynamic functional methods and
the diagrammatic expansions and resummations used to derive macroscopic
equations from the microscopic dynamics. We show why the macroscopic dynamic
equations for disordered models and those resulting from self-consistent
approximations to non-disordered ones coincide. We review some generic
properties of the slow out of equilibrium dynamics like the modifications of
FDT and their link to effective temperatures, some generic scaling forms of the
correlation functions, etc. Finally we solve a family of mean-field glassy
models. The connection between the dynamic treatment and the analysis of the
free-energy landscape is also presented. We use pedagogical examples all along
these lectures to illustrate the properties and results.
