On the theory of the Frankl problem for equations of mixed type

In 1956 Frankl, while studying subsonic flows past a profile with a supersonic zone terminating with a normal compression shock, arrived at a new mathematical problem for the Chaplygin equation with a non-local boundary condition. In this article we give a survey of classical and recent papers dedicated to this problem. We present theorems on the existence and uniqueness of the solution of the Frankl problem, study the spectral problem for the Lavrent'ev-Bitsadze operator, show applications of these results to the construction of a solution with the aid of a series, and state some unsolved problems.