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 ≤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.