Spectral statistics for random Hamiltonians in the localized regime

We present recent results on the level, level spacings and localization center statistics for random Hamiltonians in the localized regime. The main tools of the analysis are a characterization of the localized regime which can be obtained through multiscale analysis or through the fractional moment method and local spectral statistics (Wegner and Minami's estimate). These ingredients essentially allow us to realize the spectral data of random Hamiltonians as independent random objects; this description is then used to obtain the results on the statistics. If time allows, we will also explain new spectral decorrelation estimates.