We say that two algebraic varieties are birational to each other if they contain open dense subsets isomorphic to each other. Within every class of birational equivalence we can find simple varieties, which we call minimal models. Minimal models may not be unique, the projective space has many minimal models. These are either Fano varieties (varities with a positive curvature metric) or fibrations with Fano varieties as fibers. I study birationa; geometry of three-dimensional Fano fibrations over the projective line. I discuss the results concerning uniqueness of a minimal model within birational class and stability of not being birational to the projective space.