  Disordered systems submitted to a slowly increasing external stress often
reacts with a jerky dynamics characterized by bursts of activity, called
avalanches, which are the manifestation of an out-of-equilibrium phase
transition. This thesis focuses on the dynamics of elastic interfaces near the
depinning transition, and more specially on the effect of long-range
interactions. I derive the universal scaling form of the correlation functions
of the local velocity field. It allows to test the universality class of the
transition as well as the range of the interactions. I also study the
statistics of the clusters that form in presence of long-range interactions and
show how it relates to the statistics of the global avalanches. Finally I
present analytical advances towards the understanding of the spatial structure
of avalanches within a mean-field model, the Brownian force model.
