  Conformal extensions of Levy-Leblond's Carroll group, based on geometric
properties analogous to those of Newton-Cartan space-time are proposed. The
extensions are labelled by an integer $k$. This framework includes and extends
our recent study of the Bondi-Metzner-Sachs (BMS) and Newman-Unti (NU) groups.
The relation to Conformal Galilei groups is clarified. Conformal Carroll
symmetry is illustrated by "Carrollian photons". Motion both in the
Newton-Cartan and Carroll spaces may be related to that of strings in the
Bargmann space.
