  We study KPZ surfaces on Euclidean lattices and directed polymers on
hierarchical lattices subject to different distributions of disorder, showing
that universality holds, at odds with recent results on Euclidean lattices.
Moreover, we find the presence of a slow (power-law) crossover toward the
universal values of the exponents and verify that the exponent governing such
crossover is universal too. In the limit of a 1+epsilon dimensional system we
obtain both numerically and analytically that the crossover exponent is 1/2.
