Abstract: In recent years, propagation estimates known as Lieb-Robinson bounds have been obtained and refined for broad classes of quantum lattice systems. The bounds express the locality of the dynamics of quantum systems with short range interactions. Simply stated, the dynamics up to time $t>0$ of an observable involving the degrees of freedom in a bounded region of space depends in an essential way only on the degrees of freedom located at distance $d$ less than $vt$. The propagation estimates provide a bound for $v$, which is called the Lieb-Robinson velocity. This simple property has proved to be a key element in a number of interesting new results on quantum dynamics ground states. We will discuss a recent application to the characterization of gapped ground state phases of quantum spin systems. This is a joint work with Sven Bachmann, Spyridon Michalakis, and Robert Sims.