This paper gives a brief survey of recent developments on mathematical modeling and analysis of the open cavity scattering problems,which arise in diverse scientific areas and have significant industrial and military applications.The scattering problems are studied for the two-dimensional Helmholtz equation corresponding to the transverse magnetic or electric polarization,and the three-dimensional time-harmonic and time-domain Maxwell equations.Since these problems are imposed in open domains,a key step of the analysis is to develop transparent boundary conditions and reformulate them equivalently into boundary value problems in bounded domains.The well-posedness of weak solutions are shown for the associated variational problems by using either the Lax-Milgram theorem or the Fredholm altative.