mori fiber spaces (mfs) are one of the building blocks in the minimal model program. these are maps $x \to y$ between normal varieties with nice singularities, such that $\dim y <\dim x$, $\rho(x/y)=1$ and $-k_x$ is ample on every fiber. in particular, most fibers will be $q$-fano varieties. starting from classical results on the topology of fibrations, i will try to explain how the above conditions place strong restrictions on what varieties can appear as fibers of mfs. i will give characterizations for low-dimensional varities and explain what happens in the toric category. moreover, we will show that this question can be connected to the question of existence of kaehler-einstein metrics. joint work with g. codogni, a. fanelli, l. tasin.