Classical Seshadri constants are local positivity measures for line bundles around points. We extend these to arbitrary rank. We find a local to global ampleness criterion, a geometric characterization of the loci where Seshadri constants vanish, and a Seshadri interpretation to asymptotic jet separation. We also look at applications of our theory to characterizations of projective space, to a conjecture on the ample cone of the self-product of a general curve, and to a relative version of the Fujita conjecture. This is all in joint work with Takumi Murayama.