  We study the crossover from self--similar scaling behavior to asymptotically
self--affine (anisotropic) structures. As an example, we consider bond
percolation with one preferred direction. Our theory is based on a
field--theoretical representation, and takes advantage of a renormalization
group approach designed for crossover phenomena. We calculate effective
exponents for the connectivity describing the entire crossover region from
isotropic to directed percolation, and predict at which scale of the anisotropy
the crossover should occur. We emphasize the broad range of applicability of
our method.
