  General bounds are presented for the diameters of orbital graphs of finite
affine primitive permutation groups. For example, it is proved that the orbital
diameter of a finite affine primitive permutation group with a nontrivial point
stabilizer $ H \leq \mathrm{GL}(V) $, where the vector space $ V $ has
dimension $ d $ over the prime field, can be bounded in terms of $ d $ and $
\log |V| / \log |H| $ only. Several infinite families of affine primitive
permutation groups with large orbital diameter are constructed. The results are
independent from the classification of finite simple groups.
