What are some general classes of compact complex manifolds whose universal covers are bounded domains? One class I know are the Kodaira fibered surfaces.



Natural classes of compact complex manifolds whose universal covers are bounded domains
I'm not aware of a sufficient condition, although many classes have been recognized as carrying the "bounded domain" covers. Here is [one interesting paper](http://porto.polito.it/2502314/1/N50Porto.pdf) to look into.


