  We introduce a generalization of the Ozsv\'ath-Szab\'o $\tau$-invariant to
links by studying a filtered version of link grid homology. We prove that this
invariant remains unchanged under strong concordance and we show that it
produces a lower bound for the slice genus of a link. We show that this bound
is sharp for torus links and we also give an application to Legendrian link
invariants in the standard contact 3-sphere.
