  We present here the results obtained during my PhD work. We report the study
of the many body localization transition in a spin chain and the breaking of
ergodicity measured in terms of return probability in a state evolution.
Moreover, we introduce the Richardson model, an exactly solvable model, that
turns out to be suitable for investigating the many-body localized phase. Then
we turn to the analysis of the quench problem in an ensemble of random
matrices. We analyze the thermalization properties and the validity of the
Eigenstate Thermalization Hypothesis for the typical case, where the quench
parameter explicitly breaks a Z_2 symmetry.
