Yes. And local bank debit cards usually combined on the VISA card, in Norway there is BankAxept, Dankort in Denmark and something else Germany can't remember. The fees for these transactions are next to nothing compared to Visa/MC here in Norway. But very rarely accepted online (unfortunately).
This is similar in Australia and New Zealand as well. Visa and MasterCard are extremely common, AMEX is sometimes offered with fancy accounts (usually as a second card), and Discover is all but unheard of.
We have a checking account payment system called EFTPOS which has near universal acceptance, and can be combined on the same card as a Visa/MC/AMEX.
Contactless payment (PayWave and PayPass) is rapidly becoming commonplace among retailers, and can be used for transactions up to AU$100.
Australia plans to deprecate the use of signature verification for domestic credit cards later this year, requiring Chip+PIN for all cards issued by Australian banks. Signatures will still be accepted for foreign cards, because tourists.
Visa Debit is available and works, but doesn't really have any domestic advantages over EFTPOS. Maestro and PLUS aren't brands Australians generally interact with, except when traveling.
Frustratingly, Australian Chip+PIN cards don't seem to travel overseas well -- in ATMs they work fine (with fees galore) but in retail, the terminal often insists on a signature instead of a PIN.
Yes, there are often country-specific debit cards, but I'm not aware of many country-specific credit cards in Europe? Within the EU (and especially Eurozone), bank transfers are all easy to do and interoperable.