  We study the effect of generic spatial anisotropies on the scaling behavior
in the Kardar-Parisi-Zhang equation. In contrast to its "conserved" variants,
anisotropic perturbations are found to be relevant in d > 2 dimensions, leading
to rich phenomena that include novel universality classes and the possibility
of first-order phase transitions and multicritical behavior. These results
question the presumed scaling universality in the strong-coupling rough phase,
and shed further light on the connection with generalized driven diffusive
systems.
