  We present an operational framework for Einstein-Podolsky-Rosen steering as a
physical resource. To begin with, we characterize the set of steering
non-increasing operations (SNIOs) --i.e., those that do not create steering--
on arbitrary-dimensional bipartite systems composed of a quantum subsystem and
a black-box device. Next, we introduce the notion of convex steering monotones
as the fundamental axiomatic quantifiers of steering. As a convenient example
thereof, we present the relative entropy of steering. In addition, we prove
that two previously proposed quantifiers, the steerable weight and the
robustness of steering, are also convex steering monotones. To end up with, for
minimal-dimensional systems, we establish, on the one hand, necessary and
sufficient conditions for pure-state steering conversions under stochastic
SNIOs and prove, on the other hand, the non-existence of steering bits, i.e.,
measure-independent maximally steerable states from which all states can be
obtained by means of the free operations. Our findings reveal unexpected
aspects of steering and lay foundations for further resource-theory approaches,
with potential implications in Bell non-locality.
