  We introduced few years ago a new notion of causality for noncommutative
spacetimes directly related to the Dirac operator and the concept of Lorentzian
spectral triple. In this paper, we review in a non-technical way the
noncommutative causal structure of many toy models as almost commutative
spacetimes and the Moyal-Weyl spacetime. We show that those models present some
unexpected physical interpretations as a geometrical explanation of the
Zitterbewegung trembling motion of a fermion as well as some geometrical
constraints on translations and energy jumps of wave packets on the Moyal
spacetime.
