  Using the representation theory of Cherednik algebras at t=0 and a Galois
covering of the Calogero-Moser space, we define the notions of left, right and
two-sided Calogero-Moser cells for any finite complex reflection group. To each
Caloger-Moser two-sided cell is associated a Calogero-Moser family, while to
each Calogero-Moser left cell is associated a Calogero-Moser cellular
representation. We study properties of these objects and we conjecture that,
whenever the reflection group is real (i.e. is a Coxeter group), these notions
coincide with the one of Kazhdan-Lusztig left, right and two-sided cells,
Kazhdan-Lusztig families and Kazhdan-Lusztig cellular representations.
