  We develop the concept of genuine N-partite Einstein-Podolsky-Rosen (EPR)
steering. This nonlocality is the natural multipartite extension of the
original EPR paradox. Useful properties emerge that are not guaranteed for
genuine multipartite entangled states. In particular, there is a close link
with the task of one-sided device-independent quantum secret sharing. We derive
inequalities to demonstrate multipartite EPR steering for
Greenberger-Horne-Zeilinger (GHZ) and Gaussian continuous variable (CV) states
in loophole-free scenarios.
