When a cruise crosses from America to Europe, it is called transatlantic cruises because they cross over the Atlantic Ocean. There are many cruises that do not cross over the ocean, but instead have the passengers visit small islands or other places that are close together. However, there are cruiseliners that offer transatlantic trips including Royal Caribbean. The major cruiselines like Royal and Carnival do offer transatlantic cruises.
These cruises usually cost more money than the ones that only go from a small island to small island before going back to port. It is important to save more money for these transatlantic trips because they also require more time. It seems that most of them require at least a week on the ship.
There are plenty of cruises going from America to Europe; the biggest decision you'll make is where in in Europe you'd like to go and how long you'd like your cruise to be. Transatlantic cruises generally run from 7-8 nights, 9-10, and 11-13. However, there are always longer ones available, it just may take some research.
Cunard is the only cruise line that offers 7-10-day cruises. If you have the availability for a longer cruise, the options really begin to open up with Norwegian, Celebrity Cruises, Princess, Royal Caribbean and a few others. Of course, cost will increase with the number of nights selected as well as the level of service given.