  Metrics structures stemming from the Connes distance promote Moyal planes to
the status of quantum metric spaces. We discuss this aspect in the light of
recent developments, emphasizing the role of Moyal planes as representative
examples of a recently introduced notion of quantum (noncommutative) locally
compact space. We move then to the framework of Lorentzian noncommutative
geometry and we examine the possibility of defining a notion of causality on
Moyal plane, which is somewhat controversial in the area of mathematical
physics. We show the actual existence of causal relations between the elements
of a particular class of pure (coherent) states on Moyal plane with related
causal structure similar to the one of the usual Minkowski space, up to the
notion of locality.
