JoAnn,
I was very lucky that the credit card companies picked up on "unusual" activity. Some of the attempted purchases were over the internet in France and the address to be shipped to was not mine. Others were so large the merchant was instructed to call the credit card company for approval. They were asked if the purchaser was present and the person with my credit cards couldn't tell them my mother's maiden name when asked. WHEW!
They didn't try to use my ATM card.
It may sound stupid but, if you're going to be using your ATM card don't forget to make sure you have enough money in your account!! You may not be able to transfer funds between linked accounts when you're not using your home bank's system.