More precisely: are there known manifolds which are homeomorphic, but not diffeomorphic to the standard 4-torus? Are there any nice invariants distinguishing such manifolds?
Related: if such a manifold is not known, are there any restrictions on properties of such manifolds, such as curvature?
What I have so far found works only for dimension $n\geq 5$. There are of course the results by Hsiang and Shaneson in "Fake Tori, the Annulus Conjecture, and the Conjectures of Kirby", classifying "fake tori" in $n\geq 5$ (homotopy equivalent to standard tori), which were then shown to be homeomorphic, but not diffeomorphic to the standard tori, by Hsiang and Wall in "On Homotopy Tori II".
Also, if I interpret the Bieberbach theorem on flat manifolds correctly, these exotic tori cannot be flat, since this would mean they are isometric to standard tori.