You have some secret identity that you must keep private. This can be a background, such as the fact that you are a wanted man, or required to protect your loved ones and relatives. This is not just a part of your dramatic hook; you are hounded by the inquisitive, and must spend time maintaining your identity. Enemies will sometimes try to disclose your identity and use your secret against you, and you must play along with this.
You may wear a costume be trying to keep your personal life and your heroic life appart. In this case you must always change into costume before you go into action, and sometimes your real identity is at risk and must be protected.
This limitation is very "pulpy", and should be avoided or at least not be common in campaigns that are more on the realistic side.