  In this work, we study two-dimensional Galilean field theories with global
translations and anisotropic scaling symmetries. We show that such theories
have enhanced local symmetries, generated by the infinite dimensional spin-l
Galilean algebra with possible central extensions, under the assumption that
the dilation operator is diagonalizable and has a discrete and non-negative
spectrum. We study the Newton-Cartan geometry with anisotropic scaling, on
which the field theories could be defined in a covariant way. With the
well-defined Newton-Cartan geometry we establish the state-operator
correspondence in anisotropic GCFT, determine the two-point functions of
primary operators, and discuss the modular properties of the torus partition
function which allows us to derive Cardy-like formulae.
