Very loaded, complicated answer. We first have to accept that Rome (in the west) "fell" which I'm not entirely prepared to do.
That said, I'll try to a provide a simple answer. Most assimilated into the increasingly "gothic" culture that was filtering in. Boesthius is a good example of this.
Quite a few fled to the east, in particular to Constantinople. This lead to a large Latin influx (both linguistically and culturally) to the Eastern empire, which would eventually lead to the beginning of the Justinian Dynasty which featured Latin legal texts and even a (briefly successful) attempt to "reclaim" Rome.
Take a look at this to get an idea about why this is a complicated answer. There are many interpretations of what it means to be the "end" of Rome.
If you're not going to bother searching before posting an obvious question, you could at least put a little effort in and elaborate on what you already know or specific areas / groups / times you're curious about.
Thread 3 - less than a month ago
I love this sub, but I know I am not the only person who gets annoyed seeing the same boring question asked over and over and over and over. Downvote me if necessary.