  We discuss the requirements of good statistics for quantifying
non-Gaussianity in the Cosmic Microwave Background. The importance of
rotational invariance and statistical independence is stressed, but we show
that these are sometimes incompatible. It is shown that the first of these
requirements prefers a real space (or wavelet) formulation, whereas the latter
favours quantities defined in Fourier space. Bearing this in mind we decide to
be eclectic and define two new sets of statistics to quantify the level of
non-Gaussianity. Both sets make use of the concept of cumulants of a
distribution. However, one set is defined in real space, with reference to the
wavelet transform, whereas the other is defined in Fourier space. We derive a
series of properties concerning these statistics for a Gaussian random field
and show how one can relate these quantities to the higher order moments of
temperature maps. Although our frameworks lead to an infinite hierarchy of
quantities we show how cosmic variance and experimental constraints give a
natural truncation of this hierarchy. We then focus on the real space
statistics and analyse the non-Gaussian signal generated by points sources
obscured by large scale Gaussian fluctuations. We conclude by discussing the
practical implementations of these techniques.
