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I know NOTHING about TVs, but I need to get some sort of DVD player with which to play the numerous DVDs I have around the house. Ideally, it would allow me to play DVDs from North America as well as Europe.
The TV is a Samsung UE40K5100 if that helps.
Can you suggest anything?
In this day and age I'd get a Blu-Ray rather than a DVD player, especially with that 40" telly, unless you plan on never buying shinydiscs ever again. Every Blu-Ray player will play DVDs.
First out-of-the-box multi-region player I found:
https://www.amazon.co.uk/Panasonic-DMP-BD84-MultiRegion-DVD-Side-Black/dp/B01FGEWSKE
Note this is multi-region for DVDs only. If you think you might be buying Blu-Ray discs from the US in the future, you'll need to pay a bit more.
http://www.mrmdvd.com/catalog/index.php?cPath=159