  Continuous-variable systems realized in quantum optics play a major role in
quantum information processing, and it is also one of the promising candidates
for a scalable quantum computer. We introduce a resource theory for
continuous-variable systems relevant to universal quantum computation. In our
theory, easily implementable operations---Gaussian operations combined with
feed-forward---are chosen to be the free operations, making the convex hull of
the Gaussian states the natural free states. Since our free operations and free
states cannot perform universal quantum computation, genuine non-Gaussian
states---states not in the convex hull of Gaussian states---are the necessary
resource states for universal quantum computation together with free
operations. We introduce a monotone to quantify the genuine non-Gaussianity of
resource states, in analogy to the stabilizer theory. A direct application of
our resource theory is to bound the conversion rate between genuine
non-Gaussian states. Finally, we give a protocol that probabilistically
distills genuine non-Gaussianity---increases the genuine non-Gaussianity of
resource states---only using free operations and postselection on Gaussian
measurements, where our theory gives an upper bound for the distillation rate.
In particular, the same protocol allows the distillation of cubic phase states,
which enable universal quantum computation when combined with free operations.
