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.