  This paper is devoted to the study of the random displacement model on
$\R^d$. We prove that, in the weak displacement regime, Anderson and dynamical
localization holds near the bottom of the spectrum under a generic assumption
on the single site potential and a fairly general assumption on the support of
the possible displacements. This result follows from the proof of the existence
of Lifshitz tail and of a Wegner estimate for the model under scrutiny.
