I will explain some work relating to the boundedness of holomorphic foliations on algebraic surfaces using techniques from birational geometry and the minimal model program. I will then explain some applications of these ideas to some classical problems in foliation theory (e.g., can we bound the degree of an algebraic orbit of a polynomial vector field on the plane), as well as some applications to more modern problems (e.g., moduli spaces of holomorphic foliations).