The notion of a Zariski geometry has been developed in Model Theory in the search for ``logically perfect'' structures. Zariski geometries are abstract topological structures with a dimension, satisfying certain assumptions. Algebraic varieties over algebraically closed fields and compact complex manifolds are Zariski geometries and for some time it was thought that all Zariski geometries are of this kind. In fact the classical examples are only ``limit'' cases of the general pattern, where one necessarily comes to a coordinatisation by non-commutative algebras. Conversely, we show in particular that any quantum algebra at roots of unity of certain type coordinatises a geometric object which is a Zariski geometry. This includes quantum groups.