A polynomial automorphism of a group \(G\) is an automorphism \(\varphi\) such that for certain elements \(a_ 1,...,a_{r+1}\) of \(G\) and integers \(k_ 1,...,k_ r\), \(\varphi(x)=a_ 1x^{k_ 1}a_ 2x^{k_ 2}\cdots a_ rx^{k_ r}a_{r+1}\) for all \(x\in G\). The first observation (Lemma 2.1) is that the subgroup \(J(G)\) of polynomial automorphisms consists of those automorphisms of \(G\) which are sums of inner automorphisms (the “sum” being defined pointwise). Refining a result of D. Schweigert [Arch. Math. 29, 34-38 (1977; Zbl 0368.20016)], the authors prove that if \(G\) is finite, \(J(G)\) is nilpotent of class \(c-1\) if and only if \(G\) is nilpotent of class \(c\) (Theorem 3.5). Polynomial automorphisms of nilpotent groups of class 2 and 3 are described in detail (Theorems 2.3 and 4.1) and a characterization is given of groups in which the sum of any \(r\) inner automorphisms is an automorphism (Theorem 5.2).