What would the world be like if the Roman Empire never fell?

Which "fall" are we talking about here?

Do you mean if the WRE had never collapsed (leading to the rise of the Ostrogothic, Frankish, and Gepid empires, etc.), or if the formal Roman Empire had never split into the East and the West?

 

If the former, then Francia couldn't have come to dominate, and thus the HRE never would have existed, and thus the modern state of Germany wouldn't exist at all (even though the Roman Empire(s) never controlled that territory). By the end of its life, the WRE only really controlled modern-day Italy, but if it hadn't totally collapsed than we would have seen a unified Italy a thousand years before Italy actually formed...who knows what that could have done.

This also would have slowed the growth of the Eastern Roman Empire, meaning that they would be in a much weaker position overall during the Islamic Conquests - and, as such, we might have seen Ottoman-level European muslim expansion during the time of the Rashidun or Umayyad...a thousand years earlier. This would have the subsequent effect of a much stronger Caliphate, meaning that in addition to there being more muslims in the world, they probably would have expanded more. That might have led to, for example, an islamic Tibet, or even all of Iberia under Umayyad control.

After that point, so much history has been changed that it's difficult to say. A muslim Tibet would have untold consequences on the stability of the Yuan, Ming, and Qing dynasties. A muslim Anatolia in pre-Rum times would mean that the Ottoman Empire (or an equivalent) would rise much earlier and face much less resistance, leading to a massive eastern empire that Europe wasn't ready to resist. A unified Italy without an HRE would mean that West Rome would likely continue expanding north, and thus might even lead to the overextension and subsequent collapse of the WRE centuries later...but that's just guesswork really.

/r/AskReddit Thread