  In this paper, we introduce a new structure, namely, affine Szab\'o
connection. We prove that, on $2$-dimensional affine manifolds, the affine
Szab\'o structure is equivalent to one of the cyclic parallelism of the Ricci
tensor. A characterization for locally homogeneous affine Szab\'o surface is
obtained. Examples of two- and three-dimensional affine Szab\'o manifolds are
also given.
