How was the Roman empire seen and talked about during late Middle Ages/Renaissance?

So after being to the Colosseum and seeing it standing there after 2.000 years I couldn't help but wonder: What did people think of the Roman empire when they saw it? Surely if it's standing there today it was standing back then in 1.000-1.500 AD.

What did the monarchies of the other kindgoms say when they visited Rome? Was the Roman empire talked about in the way we talk about it today? Did they know about the gladiators and the legions exploits and so on?