  We investigate the Kardar--Parisi--Zhang (KPZ) equation in $d$ spatial
dimensions with Gaussian spatially long--range correlated noise ---
characterized by its second moment $R(\vec{x}-\vec{x}') \propto
|\vec{x}-\vec{x}'|^{2\rho-d}$ --- by means of dynamic field theory and the
renormalization group. Using a stochastic Cole--Hopf transformation we derive
{\em exact} exponents and scaling functions for the roughening transition and
the smooth phase above the lower critical dimension $d_c = 2 (1+\rho)$. Below
the lower critical dimension, there is a line $\rho_*(d)$ marking the stability
boundary between the short-range and long-range noise fixed points. For $\rho
\geq \rho_*(d)$, the general structure of the renormalization-group equations
fixes the values of the dynamic and roughness exponents exactly, whereas above
$\rho_*(d)$, one has to rely on some perturbational techniques. We discuss the
location of this stability boundary $\rho_* (d)$ in light of the exact results
derived in this paper, and from results known in the literature. In particular,
we conjecture that there might be two qualitatively different strong-coupling
phases above and below the lower critical dimension, respectively.
