UNSCREW-UMS™ (U.S. Patent # 4,688,315) are designed to back out broken screws and remove other fasteners with minimal damage to the surrounding wood. Often the broken screw can be replaced with the same-sized screw; however, a longer and/or thicker screw may be needed, especially if structural strength is important. Occasionally, the hole must be epoxied or plugged and redrilled. The important thing is that something can be done because the old screw is gone.
Article in Fine Woodworking Magazine