A foliation on an algebraic variety is a partition of the variety into ``parallel'' disjoint immersed complex submanifolds. \\ \\ This turns out to be a very useful notion and holomorphic foliations have played a central role in several recent developments in the study of the geometry of projective varieties. \\ \\ This is the first part of a two talks series (with Roberto Svaldi) in which we will explain some recent work building towards the birational classification of holomorphic foliations on projective varieties in the spirit of the Minimal Model program. We will explain some applications of these ideas to the study of the dynamics and geometry of foliations and foliation singularities. \\ \\ Features joint work with P. Cascini and R. Svaldi