The Minimal Model Program aims to classify all algebraic varieties X (up to birational equivalence) via three building blocks: Fano varieties, Calabi-Yau varieties and canonically polarized varieties. Given a fibration from X to Z, for inductive reasons, we want to induce canonically certain structures on both X and Z associated to the fibration. It turns out that the correct structures are foliations and generalized pairs. The latter were formally introduced by Birkar and Zhang. I will talk about some recent progress in the Minimal Model Program for both generalized pairs and foliations. This talk is based on a series of joint works with G.Chen, J. Han, J. Liu and N.Tsakanikas.