As a consequence of Minimal Model Program, a rationally connected
variety is birational to a suitable fibration into Q-Fano varieties
which is called a Mori fiber space. A birational Mori fiber structure
of an algebraic variety is a Mori fiber space birational to the given
variety. I will talk about the study of birational Mori fiber
structures of Q-Fano weighted complete intersections. This is deeply
related with the rationality problem of algebraic varieties.