Given an undirected simple graph G=(V,E) and integer values f(v) for each node v in V, a node subset D is called an f-tuple dominating set if, for each node v in V, its closed neighborhood intersects D in at least f(v) nodes. We investigate the polyhedral structure of the polytope that is defined as the convex hull of the incidence vectors of the f-tuple dominating sets in G. We provide a complete formulation for the case of stars and introduce a new family of (generally exponentially many) inequalities which are valid for the f-tuple dominating set polytope of any graph. A corollary of our results is a proof that a conjecture present in the literature on a complete formulation of the 2-tuple dominating set polytope of trees does not hold. Investigations on adjacency properties in the 1-skeleton of the f-tuple dominating set polytope are also reported