They do not have a surname as part of their legal names, but they are entitled to the use of other surnames. Legally the direct descendants of Queen Elizabeth II can use "Windsor." Others in the family not descended from the Queen can use "Mountbatten-Windsor." Or Saxe-Coburg, I think. However, many of the people entitled to use "Windsor" choose to use "Muntbatten-Windsor" out of respect for Prince Philip.
Also, they can use placeholder surnames. Prince Harry uses "Wales."