No, British Shorthairs do not run away. They are a very relaxed and easygoing breed of cat, content to stay close to home. However, they may wander off if they get bored or if there is something interesting going on nearby that catches their attention. If you have an indoor British Shorthair, it is important to provide them with plenty of stimulation in the form of toys, games and human interaction to keep them from getting restless.