Summary: Let \(G = (V, E)\) be an undirected graph where the edges in \(E\) have non-negative weights. A star in \(G\) is either a single node of \(G\) or a subgraph of \(G\) where all the edges share one common end-node. A star forest is a collection of vertex-disjoint stars in \(G\). The weight of a star forest is the sum of the weights of its edges. This paper deals with the problem of finding a Maximum Weight Spanning Star Forest (MWSFP) in \(G\). This problem is NP-hard but can be solved in polynomial time when \(G\) is a cactus [V. H. Nguyen, Discrete Math. Algorithms Appl. 7, No. 2, Article ID 1550018, 10 p. (2015; Zbl 1316.05095)]. In this paper, we present a polyhedral investigation of the MWSFP. More precisely, we study the facial structure of the star forest polytope, denoted by \(SFP(G)\), which is the convex hull of the incidence vectors of the star forests of \(G\). First, we prove several basic properties of \(SFP(G)\) and propose an integer programming formulation for MWSFP. Then, we give a class of facet-defining inequalities, called M-tree inequalities, for \(SFP(G)\). We show that for the case when \(G\) is a tree, the \(M\)-tree and the nonnegativity inequalities give a complete characterization of \(SFP(G)\). Finally, based on the description of the dominating set polytope on cycles given by M. Bouchakour et al. [Eur. J. Comb. 29, No. 3, 652–661 (2008; Zbl 1168.05340)], we give a complete linear description of \(SFP(G)\) when \(G\) is a cycle.