Consider a connected homogeneous Riemannian manifold \((M,ds^2)\) and a Riemannian covering \((M,ds^2) \to \Gamma \backslash (M,ds^2)\) . If \(\Gamma \backslash (M,ds^2)\) is homogeneous, then every \(\gamma \in \Gamma \) is an isometry of constant displacement. The homogeneity conjecture suggests the converse: if every \(\gamma \in \Gamma \) is an isometry of constant displacement on \((M,ds^2)\) , then \(\Gamma \backslash (M,ds^2)\) is homogeneous. We survey the cases in which the homogeneity conjecture has been verified, including some new results, and suggest some related open problems.