This is an invitation to explore the `quantum' geometry.
The starting point will be a formulation of such structures
on a `classical' space X like group, topological,
differential, Riemannian or spin,
in terms of (commuting) algebra of functions F(X).
Next a generalization to noncommutative algebras will be indicated
and illustrated by a number of selected examples
which are of interest in mathematical physics
and link to some open research problems.