Backpacking Europe for 2 months, should I get Round or One-way

Your best approach would be an open jaw ticket - fly into one city and home from another. For example fly into London and come home from Munich.

I agree with Frank. An open-jaw ticket would be a good idea, and that way you'll be able to demonstrate that you have a ticket to return home. If you know you'll be travelling for two months, I'm not sure how much "flexibility" you'll need.

Generally, trans-Atlantic tickets one-way are very costly, maybe even as much as a return ticket. Train travel or inter-Europe flights are priced one-way as a matter of course. But it is costly to wait until the last minute for some train and most airline purchases, since tickets can be far cheaper if bought well in advance. The suggestion to use a multi-city plan is a practical one, employing the multi-destination search function on many travel agency and airline websites. That will pin you down to both a date and a departure city, but on some airlines those may be altered, at a considerable fee. Flexibility is expensive.

What you are looking for is the "open-jaw" ticket. When looking at flights, choose the "multiple destination" option (sometimes it's "return from another location" or similar).

The best way to start is to decide which places you want to visit, then work out a logical (more-or-less straight) path from the places that are farthest apart. Starting in May, you may want to go from south to north, avoiding the heat of early summer.

But also consider cheap flights within Europe. It might work out better to get a round trip ticket. For instance, you might get a good deal on a round-trip ticket to Dublin, then get a cheap flight to Athens and work your way back across Europe. Depends on where you're flying from and where you want to go in Europe.