  We propose a measure of non-Gaussianity for quantum states of a system of $n$
oscillator modes. Our measure is based on the quasi-probability $Q(\alpha),
\alpha\in{\cal C}^n$.
  Since any measure of non-Gaussianity is necessarily an attempt at making a
quantitative statement on the departure of the shape of the $Q$ function from
Gaussian, any good measure of non-Gaussianity should be invariant under
transformations which do not alter the shape of the $Q$ functions, namely
displacements, passage through passive linear systems, and uniform scaling of
all the phase space variables: $Q(\alpha)\to \lambda^{2n}Q(\lambda \alpha)$.
Our measure which meets this `shape criterion' is computed for a few families
of states, and the results are contrasted with existing measures of
non-Gaussianity. The shape criterion implies, in particular, that the
non-Gaussianity of the photon-added thermal states should be independent of
temperature.
