  We show that generic kinetic growth processes with surface relaxations can
exhibit a new crumpled phase with short-range orientational order at dimensions
$d<4$. A sufficiently strong spatially non-local part of the chemical potential
associated with the particle current above a threshold in the system can
trigger this crumpling. The system can also be in a perturbatively accessible
rough phase with long range orientational order but short range positional
order at $d<4$ with known scaling exponents. Intriguingly, in $d>4$ we argue
that there is no crumpling transition; instead, there is a roughening
transition from a smooth to a rough phase for large enough non-local particle
current. Experimental and theoretical implications of these results are
discussed.
