  We review the basic hypotheses which motivate the statistical framework used
to analyze the cosmic microwave background, and how that framework can be
enlarged as we relax those hypotheses. In particular, we try to separate as
much as possible the questions of gaussianity, homogeneity and isotropy from
each other. We focus both on isotropic estimators of non-gaussianity as well as
statistically anisotropic estimators of gaussianity, giving particular emphasis
on their signatures and the enhanced "cosmic variances" that become
increasingly important as our putative Universe becomes less symmetric. After
reviewing the formalism behind some simple model-independent tests, we discuss
how these tests can be applied to CMB data when searching for large scale
"anomalies"
