Ball lightning has until this time defied physical explanation. This report summarizes the requirements that must be satisfied by any valid theory, reviews and evaluates previous theories, and evaluates the new theory of Finkelstein and Rubenstein developed under an AFCRL-monitored grant. It is concluded that the latter theory is the only promising one to date but that even this theory needs to be modified. A modification is proposed which would remove the remaining objections and it is suggested that experiments rather than further theoretical development will constitute the next decisive step. Previous theories discussed are combustion, confined plasma, dust, electromagnetic radiation, cosmic ray, and quantum mechanics.