  The extension of strongly anisotropic or dynamical scaling to local scale
invariance is investigated. For the special case of an anisotropy or dynamical
exponent $\theta=z=2$, the group of local scale transformation considered is
the Schr\"odinger group, which can be obtained as the non-relativistic limit of
the conformal group. The requirement of Schr\"odinger invariance determines the
two-point function in the bulk and reduces the three-point function to a
scaling form of a single variable. Scaling forms are also derived for the
two-point function close to a free surface which can be either space-like or
time-like. These results are reproduced in several exactly solvable statistical
systems, namely the kinetic Ising model with Glauber dynamics, lattice
diffusion, Lifshitz points in the spherical model and critical dynamics of the
spherical model with a non-conserved order parameter. For generic values of
$\theta$, evidence from higher order Lifshitz points in the spherical model and
from directed percolation suggests a simple scaling form of the two-point
function.
