  We discuss the problem of characterizing the property of a Grothendieck topos
to satisfy a given 'geometric' invariant as a property of its sites of
definition, and indicate a set of general techniques for establishing such
criteria. We then apply our methodologies to specific invariants, notably
including the property of a Grothendieck topos to be localic (resp. atomic,
locally connected, equivalent to a presheaf topos), obtaining explicit site
characterizations for them.
