We shall review the attempts to extend the quantum
mechanical property of non-commutativity from phase space to
ordinary space. These attempts took a more precise form in the
case of gauge theories for which some concrete results have
been obtained. In flat space they amount to a reformulation of
the theory which looks interesting but they have not given so
far any novel physical results. However, the introduction of
gravity gives a richer structure and may offer some new
insights.