Suppose that \(X\) is a compact metric space, that \(T:X \rightarrow X\) is continuous, and that \(f\) is a continuous map from \(X\) into some Euclidean space. The corresponding generalized rotation set is the convex subset of Euclidean space consisting of all integrals of \(f\) with respect to \(T\)-invariant Borel probability measures. Properties of this set are described, usually under the assumption that the map assigning an invariant measure its entropy is upper semicontinuous. Applications to subshifts of finite type are developed. Given \(\rho\) in the rotation set, \(H(\rho)\) is defined to be the supremum of the entropies of all invariant measures \(\mu\) such that \(\int f d\mu = \rho\). This is compared with the directional entropy \(\mathcal{H}(\rho)\) recently studied by W. Geller and M. Misiurewicz [ibid. 351, 2927-2948 (1999; Zbl 0918.54019)]. It is always the case that \({\mathcal H}\leq H\); for \((X,T)\) a mixing subshift of finite type and \(f\) of summable variation, if the rotation set is strictly convex then \({\mathcal H}=H\). Without strict convexity, \(\mathcal{H}\) and \(H\) can differ only at non-exposed boundary points of the rotation set. Examples are given to show that the two either may or may not actually differ at such points.