Let G be a group and let n be a positive integer. A polynomial function in G is a function from Gn to G of the form \({(t_{1},\ldots,t_{n})\mapsto f(t_{1},\ldots,t_{n})}\), where f(x1, . . . , xn) is an element of the free product of G and the free group of rank n freely generated by x1, . . . , xn. There is a natural definition for the product of two polynomial functions; equipped with this operation, the set \({\overline{G}[x_{1},\ldots,x_{n}]}\) of polynomial functions is a group. We prove that this group is polycyclic if and only if G is finitely generated, soluble, and nilpotent-by-finite. In particular, if the group of polynomial functions is polycyclic, then necessarily it is nilpotent-by-finite. Furthermore, we prove that G itself is polycyclic if and only if the subgroup of polynomial functions which send (1, . . . , 1) to 1 is finitely generated and soluble.