  We address the quantification of non-Gaussianity of states and operations in
continuous-variable systems and its use in quantum information. We start by
illustrating in details the properties and the relationships of two recently
proposed measures of non-Gaussianity based on the Hilbert-Schmidt (HS) distance
and the quantum relative entropy (QRE) between the state under examination and
a reference Gaussian state. We then evaluate the non-Gaussianities of several
families of non-Gaussian quantum states and show that the two measures have the
same basic properties and also share the same qualitative behaviour on most of
the examples taken into account. However, we also show that they introduce a
different relation of order, i.e. they are not strictly monotone each other. We
exploit the non-Gaussianity measures for states in order to introduce a measure
of non-Gaussianity for quantum operations, to assess Gaussification and
de-Gaussification protocols, and to investigate in details the role played by
non-Gaussianity in entanglement distillation protocols. Besides, we exploit the
QRE-based non-Gaussianity measure to provide new insight on the extremality of
Gaussian states for some entropic quantities such as conditional entropy,
mutual information and the Holevo bound. We also deal with parameter estimation
and present a theorem connecting the QRE nonG to the quantum Fisher
information. Finally, since evaluation of the QRE nonG measure requires the
knowledge of the full density matrix, we derive some {\em experimentally
friendly} lower bounds to nonG for some class of states and by considering the
possibility to perform on the states only certain efficient or inefficient
measurements.
