  We present a short introductory overview of the non-commutative extensions of
several classical physical theories. After a general discussion of the reasons
that suggest that the non-commutativity is a major issue that will eventually
lead to the unification of gravity with other fundamental interactions, we
display examples of non-commutative generalizations of known geometries.
Finally we discuss the general properties of the algebras that could become
generalizations of algebras of smooth functions on Minkowskian (Riemannian)
manifolds, needed for the description of Quantum Gravity.
