We review some functional definitions of polynomial maps on abelian groups, present a modified version of Leibman's group-theoretic generalizations of the difference calculus for polynomial maps from nonempty commutative semigroups to groups, and prove that it has many desirable formal properties when the target group is locally nilpotent and also when the semigroup is the set of nonnegative integers. This is based on a work recently accepted by Journal of Group Theory.