  This thesis presents several aspects of the physics of disordered elastic
systems and of the analytical methods used for their study. On one hand we will
be interested in universal properties of avalanche processes in the statics and
dynamics (at the depinning transition) of elastic interfaces of arbitrary
dimension in disordered media at zero temperature. To study these questions we
will use the functional renormalization group. After a review of these aspects
we will more particularly present the results obtained during the thesis on (i)
the spatial structure of avalanches and (ii) the correlations between
avalanches. On the other hand we will be interested in static properties of
directed polymers in $1+1$ dimension, and in particular in observables related
to the KPZ universality class. In this context the study of exactly solvable
models has recently led to important progress. After a review of these aspects
we will be more particularly interested in exactly solvable models of directed
polymer on the square lattice and present the results obtained during the
thesis in this direction: (i) classification of Bethe ansatz exactly solvable
models of directed polymer at finite temperature on the square lattice; (ii)
KPZ universality for the Log-Gamma and Inverse-Beta models; (iii) KPZ
universality and non-universality for the Beta model; (iv) stationary measures
of the Inverse-Beta model and of related zero temperature models.
