Double posting for fun and profit.
I've got an idea for an alternate history now. The idea is that Rome accepted a ceasefire with Carthage after Cannae, and Carthage was able to dominate the sea lanes. This prevented Rome from expanding East beyond Byzantion, and the events that led to the formation of the Empire and the subsequent split between East and West never happened. Carthage, on the other hand, continued its exploration of the Atlantic, eventually traveling up the Niger river and founding colonies in sub-saharan Africa.
Meanwhile, the collapse of the Seleucid Empire allowed the Persians to gobble up the former Hellensitic lands, including Egypt and Anatolia. They share borders with both Rome and Carthage, but are maintaining a very tenuous peace with them. Since the Persians were not weakened by wars with Rome, the Arabs were not able to launch their conquests, although their religion spread all rapidly the same, and Mecca is still a very important city.
Finally, in the 5th century, Rome managed to repel the Huns and barbarian hordes, and has survived to the current date of 750 AD, albeit not without heavy sacrifice; they lost most of their northern territories, particularly Britain, which the Celts managed to reacquire. But the soon-to-commence Viking raids will throw the north into even more chaos, and may allow Rome the chance to retake its former lands.
Asia, on the other hand, is largely dominated by the Tang Dynasty in China and the Pala Empire in India, much as was in real life. This is largely because I don't know much about Asian history and couldn't think of anything interesting for them.
Now, this whole situation, as you might have inferred, is a bit of a stretch. It's not likely that a Carthaginian peace would have lasted beyond Hannibal's lifetime, much less any of the other stuff I mentioned. But still, it's the best I can do.
I've already begun work on it.