  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.
