Based on my recent experience, Discover is spotty in Europe but some places definitely accept it. Not that great if you're on vacation and want to avoid hassle, though. Discover is my main driver (both credit and bank) in the US, but I decided to use a Visa card in Europe just to avoid random rejections. But you won't be screwed in western Europe if all you have is a Discover card.

Really depends on the country. Discover acceptance is actually relatively good globally, as they have interconnections to Diners Club, JCB and China UnionPay.