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Abstract. We dene a polycyclic restricted Lie algebra to be the Lie analog of a polycyclic group, and we describe the structure of poly(cyclic or nite-dimensional) restricted Lie algebras. In partic-ular, we prove that these are precisely the restricted Lie algebras whose restricted enveloping algebras have polynomial growth. 1. Polycyclic restricted Lie algebras Denote the ground eld by K and let char K = p> 0. We refer the reader to [4] for the denition and basic properties of a restricted Lie algebra. Let L be such an algebra over the eld K. If the K-subspace H L is closed under the Lie bracket, then H is said to be a Lie subalgebra of L. In this case, one denes its p-hull to be Hp = hhpn j h 2 H; n 0iK. Properties of the p-mapping imply that Hp is the minimal restricted subalgebra containing H, and that [Hp; Hp] = [H; H]. We say that a restricted Lie algebra L is cyclic if it is generated by one element e as a p-algebra, that is L = Hp where H = heiK. It follows from the above that L is abelian, and there are two possibilities. First, we say that e is algebraic with respect to the p-mapping if there exists an integer n such that e