Why should humanity 'stop' everything as it sees fit with imperfect knowledge and burgeoning ego? Cockroaches have a valuable role to play in degrading the vast quantities of biological waste we put out. We don't need to stop them.
Here in DC, we get them, but generally only in homes that aren't kept clean - which can a problem in large apartment buildings, as it only takes one slovenly neighbor or poorly tended garbage area to let a colony grow and become a visible nuisance.
And what about Europe? Probably too cold in most of the continent, but present in Mediterranean regions?
Seems like a U.S exclusive bug.
As with other areas they live in or near garbage areas, but as far as I can tell not as common as one could think. I don't recall seeing any for example during garbage collecting emergencies. They could be present in homes though, especially old ones where small cracks in concrete or wood can offer a place to hide. That was the case of a 4th floor apartment I lived in over 30 years back where we were never able to find the source of those bastards. That wasn't an invasion, just 2 or 3 a week, so we weren't that pressured to call for someone to eradicate them. The solution came totally unexpected: one day my father brought a cat into that house and from that moment although we keep finding those 2 or 3 cockroaches per week, they were dead. Some time passed by, and one day I found the cat scratching near a baseboard corner; he just found the small crack the roaches used to live in and killed all those coming out of it. We sealed the crack and never saw a cockroach after that. I moved two times since then, and although my father kept that cat, I soon got two, and besides being wonderful companions, they kill anything entering the house uninvited, including bigger nasty mosquitos.
Friends at university said they have plenty in dorms, but a visit made it self-explanatory.
The main problem is that, like rodents, they can be a major health hazard as passive disease carriers. A pet rat is as different from a wild rat as a pet roach is as different from a wild roach.