  We develop a local positivity theory for movable curves on projective
varieties similar to the classical Seshadri constants of nef divisors. We give
analogues of the Seshadri ampleness criterion, of a characterization of the
augmented base locus of a big and nef divisor, and of the interpretation of
Seshadri constants as an asymptotic measure of jet separation. We also study
the case of arbitrary codimension.
