  We investigate structure functions in the 2--dimensional (asymptotically
free) non--linear O$(n) \sigma$--models using the non--perturbative S--matrix
bootstrap program. In particular the {\it exact small (Bjorken) $x$ behavior is
exhibited}; the structure is rather universal and is probably the same in a
wide class of (integrable) asymptotically free models. Structurally similar
universal formulae may also hold for the small $x$ behavior of QCD in
4--dimensions. Structure functions in the special case of the $n=3$ model are
accurately computed over the whole $x$ range for $-q^2/M^2<10^5$, and some
moments are compared with results from renormalized perturbation theory. Some
remarks concerning the structure functions in the $1/n$ approximation are also
made.
