In this paper, we study polynomial structures by starting on the Lie algebra level, then passing to Lie groups to finally arrive at the polycyclic-by-finite group level. To be more precise, we first show how a general solvable Lie algebra can be decomposed into a sum of two nilpotent subalgebras. Using this result, we construct, for any simply connected, connected solvable Lie group G of dim n, a simply transitive action on Rn which is polynomial and of degree ≤ n3. Finally, we show the existence of a polynomial structure on any polycyclic-by-finite group Γ, which is of degree ≤ h(Γ)3 on almost the entire group (h (Γ) being the Hirsch length of Γ).