Higher dimensional riemann mapping theorem.

Question

Riemann mapping theorem states that any simply connected domain in $\mathbb{C}$ is conformally diffeomorphic to unit disc. Is it true in higher dimensions? I guess the answer is no. I read a few papers which say that only a few regions in higher dimensions are conformally diffeomorphic to the unit ball. I think the reason might be due to Liouville's theorem that conformal mappings are just Mobius transformations in higher dimensions, so it's quite restricted. But I am still curious about those examples that are conformally diffeomorphic to the unit ball, are they only the images of the unit ball under random Mobius transformations? The reason I am thinking about this is that I want to see a conformal diffeomorphism between the unit ball and a smooth convex body, I think in general there is no such map.