  We address the issue of quantifying the non-Gaussian character of a bosonic
quantum state and introduce a non-Gaussianity measure based on the
Hilbert-Schmidt distance between the state under examination and a reference
Gaussian state. We analyze in details the properties of the proposed measure
and exploit it to evaluate the non-Gaussianity of some relevant single- and
multi-mode quantum states. The evolution of non-Gaussianity is also analyzed
for quantum states undergoing the processes of Gaussification by loss and
de-Gaussification by photon-subtraction. The suggested measure is easily
computable for any state of a bosonic system and allows to define a
corresponding measure for the non-Gaussian character of a quantum operation.
