If my debit or credit card is stolen and used within the EU, am I liable for any of the bill or is it paid for by the credit card provider (assuming that I have not been negligent)?

The most you will ever have to pay if you are the innocent victim of card fraud is 50. In practice, most banks refund you with the full amount. However, if your card issuer can show that you have acted fraudulently or without reasonable care, you may be liable for all losses on the card. A common example of acting without reasonable care is writing down your PIN and storing it with your card.

A full explanation of liability for losses can be found in sections 12.11 and 12.12 of The Banking Code.