In this thesis we address several questions related to important conjectures in birational geometry. In the first two chapters we prove that it is possible to bound the number of minimal models of a smooth threefold of general type depending on the topology of the underlying complex manifold. Moreover, under some technical assumptions, we provide some explicit bounds and we explain the relationship with the effective version of the finite generation of the canonical ring. Then we prove the existence of rational curves on certain type of fibered Calabi-Yau manifolds. Finally, in the last chapter we move to birational geometry in positive characteristic and we prove the Base point free Theorem for a three dimensional log canonical pair over the algebraic closure of a finite field.