For some date pairs in September, you can fly WOW Airlines from Los Angeles LAX to Edinburgh, Scotland EDI for just $519 round-trip.
The WOW Air website is easy to use. You can find it at http://wowair.us/
WOW is a very reputable airline based in Iceland, with many destinations in Europe. WOW trans-Atlantic flights stop in Iceland.
WOW charges for many things that are free on other airlines, including water, so be sure to take an empty water bottle with you through security and fill it before approaching the gate.