  We develop a general framework to assess capabilities and limitations of the
Gaussian toolbox in continuous variable quantum information theory. Our
framework allows us to characterize the structure and properties of quantum
resource theories specialized to Gaussian states and Gaussian operations,
establishing rigorous methods for their description and yielding a unified
approach to their quantification. We show in particular that, under a few
intuitive and physically motivated assumptions on the set of free states, no
Gaussian quantum resource can be distilled with free Gaussian operations, even
when an unlimited supply of the resource state is available. This places
fundamental constraints on state manipulations in all such Gaussian resource
theories. We discuss in particular the applications to quantum entanglement,
where we extend previously known results by showing that Gaussian entanglement
cannot be distilled even with Gaussian operations preserving the positivity of
the partial transpose, as well as to other Gaussian resources such as steering
and optical nonclassicality. A comprehensive semidefinite programming
representation of all these resources is explicitly provided.
