We discuss localization and local eigenvalue statistics for Schr\"odinger operators with random point interactions on $R^d$, for $d=1,2,3$. The results rely on probabilistic estimates, such as the Wegner and Minami estimate, for the eigenvalues of the Schr\"odinger operator restricted to cubes. The special structure of the point interactions facilitates the proofs of these eigenvalue correlation estimates.
One of the main results is that the local eigenvalue statistics is given by a Poisson point process in the localization regime, one of the first examples of Poisson eigenvalue statistics for multi-dimensional random Schr\"odinger operators in the continuum.  This is joint work with M.\ Krishna and W.\ Kirsch.