  We construct classical theories for scalar fields in arbitrary Carroll
spacetimes that are invariant under Carrollian diffeomorphisms and Weyl
transformations. When the local symmetries are gauge fixed these theories
become Carrollian conformal field theories. We show that generically there are
at least two types of such theories: one in which only time derivatives of the
fields appear and the other in which both space and time derivatives appear. A
classification of such scalar field theories in three (and higher) dimensions
up to two derivative order is provided. We show that only a special case of our
theories arises in the ultra-relativistic limit of a covariant parent theory.
