Your question is to broad (where in USA? where in Europe? When?) but if you don’t put any restrictions (anywhere in EU, anywhere in USA, in the next two month) these are some results:
It makes clear that Norwegian is your best option.
How I find these rates
Enter SkyScanner and write:
You must repeat this procedure with every European country.
This is really a tricky problem.
XL has reasonably cheap fares from Paris to NY, one way.
(So, travel in summer 2015, about 500 euros .. compare roughly 2000 euros for any other major airline option.) Hope it helps.
Disclaimer: I actually never booked a single flight through them.
Rome2Rio gives the following prices for single routes:
I didn’t enter specific dates.
Air Transat offers flights to Canada (Paris-Montreal can be quite cheap out of summer, even few days before booking).
Icelandair to North America via Reykjavik (they even offer the possibility for a stopover in Iceland if you want to visit). Icelandair have specific “gateways” on each side of the Ocean, airports for which they have more frequent and cheaper flights.
Both airlines have reasonable prices for their One-Way flights (still not much less than 400 euros)
I also once flew over the ocean booking via Hipmunk and the deal was great, but I don’t know how often that happens.
Credit:stackoverflow.com‘
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