And from across the pond comes............ The Long Good Friday.
It does have a Mafia family in it, although they don't mention their name.

What about Miller's Crossing, Billy Bathgate (sorry just noticed it was mentioned) , also Once Upon A Time In America, I know it has been mentioned but this is a Classic, so it deserves another.

You've got me thinking now.... The Last Capone, not 100% on the title but it's about an illegitimate son of Al. Really crap, my brother saw the title and bought me it a few years ago.

Cotton Club.... a semi musical Mafia film.. it's actually quite good.