Are you looking for problems with specific models about how the universe evolved following the Big Bang? There are some open questions on most of these, which I'd be happy to provide if you want to talk about them.
On the other hand, if you're looking for problems with the conclusion that the universe came from a high-temperature, high-density state in the distant past (essentially that there was a Big Bang), then there genuinely isn't really any open problems with that idea.