  Gaussian states have played on important role in the physics of
continuous-variable quantum systems. They are appealing for the experimental
ease with which they can be produced, and for their compact and elegant
mathematical description. Nevertheless, many proposed quantum technologies
require us to go beyond the realm of Gaussian states and introduce non-Gaussian
elements. In this Tutorial, we provide a roadmap for the physics of
non-Gaussian quantum states. We introduce the phase-space representations as a
framework to describe the different properties of quantum states in
continuous-variable systems. We then use this framework in various ways to
introduce extra structure in the state space. We explain how non-Gaussian
states can be characterised not only through the negative values of their
Wigner function, but also via other properties such as quantum non-Gaussianity
and the related stellar rank. For multimode systems, we are naturally
confronted with the question of how non-Gaussian properties behave with respect
to quantum correlations. To answer this question, we first show how
non-Gaussian states can be created by performing measurements on a subset of
modes in a Gaussian state. Then, we highlight that these measured modes must be
correlated via specific quantum correlations to the remainder of the system to
create quantum non-Gaussian or Wigner-negative states. On the other hand,
non-Gaussian operations are also shown to enhance or even create quantum
correlations. Finally, we will demonstrate that Wigner negativity is a
requirement to violate Bell inequalities and to achieve a quantum computational
advantage. At the end of the Tutorial, we also provide an overview of several
experimental realisations of non-Gaussian quantum states in quantum optics and
beyond.
