Covering space and classification Is there any classification for all topological spaces that have $\mathbb{R}^n$ as their universal cover?

A more restrictive question has a more reasonable answer, namely a classification of all Riemannian manifolds that have $\mathbb{R}^n$, with the usual metric, as their universal cover. I think that these are precisely the complete flat manifolds. Bieberbach classified all the closed flat manifolds: in particular, they all admit finite covers by flat tori $\mathbb{R}^n/\Gamma$ (where $\Gamma$ is a lattice). Their fundamental groups are space groups. 
As Mike Miller says in the comments, without this restriction the problem is at least as complicated as the problem of classifying hyperbolic manifolds (since hyperbolic space is homeomorphic and even diffeomorphic to Euclidean space), which is probably hopeless. 
