Locality in quantum spin systems can be described in terms of a bound onthe commutator of local observables with disjoint supports. The first proof of such estimates was given by Lieb and Robinson in 1972. In mytalk, I will discuss a generalization of these results and a variety ofapplications. For example, these new results imply the existence of thedynamics for a wider class of interactions, a bound on the propagation ofcorrelations through the system, and a proof of exponential decay of (spatial)correlations in the ground state of gapped Hamiltonians. Moreover,combining the above mentioned applications, we were recently able to provea multi-dimensionalLieb-Schultz-Mattis theorem concerning the existence of low-lying excitations for thespin-1/2 antiferromagnetic Heisenberg model.