  This review reports on the research done during the past years on violations
of the fluctuation-dissipation theorem (FDT) in glassy systems. It is focused
on the existence of a quasi-fluctuation-dissipation theorem (QFDT) in glassy
systems and the currently supporting knowledge gained from numerical simulation
studies. It covers a broad range of non-stationary aging and stationary driven
systems such as structural-glasses, spin-glasses, coarsening systems,
ferromagnetic models at criticality, trap models, models with entropy barriers,
kinetically constrained models, sheared systems and granular media. The review
is divided into four main parts: 1) An introductory section explaining basic
notions related to the existence of the FDT in equilibrium and its possible
extension to the glassy regime (QFDT), 2) A description of the basic analytical
tools and results derived in the framework of some exactly solvable models, 3)
A detailed report of the current evidence in favour of the QFDT and 4) A brief
digression on the experimental evidence in its favour. This review is intended
for inexpert readers who want to learn about the basic notions and concepts
related to the existence of the QFDT as well as for the more expert readers who
may be interested in more specific results.
