It is well-known that classification of manifolds up to homemorphism is, in general, out of question. However, this task is sometimes tractable under some additional assumptions on manifolds one would like to classify. I want to ask about one specific example of this situation.

Let $T^n$ denote the $n$-dimensional torus $(S^1)^n$. Suppose that a closed manifold is finitely covered by $T^n$. I would like to know to what extent (and whether at all) it is possible to classify such manifolds $M$. If $n=1$, then $M=S^1$; if $n=2$, then $M=S^1 \times S^1$ or $M =K$, the Klein bottle. (But then again, this is not really interesting, because in these dimensions all manifolds are classified.) What happens if $n \geq 3$, or at least if $n=3$?