  Glassy systems are disordered systems characterized by extremely slow
dynamics. Examples are supercooled liquids, whose dynamics slow down under
cooling. The specific pattern of slowing-down depends on the material
considered. This dependence is poorly understood, in particular, it remains
generally unclear which aspects of the microscopic structures control the
dynamics and other macroscopic properties. Attacking this question is one of
the two main aspects of this dissertation. We have introduced a new class of
models of supercooled liquids, which captures the central aspects of the
correspondence between structure and elasticity on the one hand, the
correlation of structure and thermodynamic and dynamic properties on the other.
Our results shed new light on the temperature-dependence of the topology of
covalent networks, in particular, on the rigidity transition that occurs when
the valence is increased.
  Other questions appear in glassy systems at zero temperature. In that
situation, a glassy system can flow if an external driving force is imposed
above some threshold. The first example we will consider is the erosion of a
riverbed. Experiments support the existence of a threshold forcing, below which
no erosion flux is observed. In this dissertation, we present a novel
microscopic model to describe the erosion near threshold. This model makes new
quantitative predictions for the spatial reparation of the flux. To study
further the self-organization of driven glassy systems, we investigate the
athermal dynamics of mean-field spin glasses. The spin glass self-organizes
into the configurations that are stable, but barely so. Such marginal stability
appears with the presence of a pseudogap in soft excitations. We show that the
emergence of a pseudogap is deeply related to very strong anti-correlations
emerging among soft excitations.
