Foliations are important and connect many areas of mathematics including algebraic geometry, complex analysis, topology, etc. Due to the close connection between foliations and the open conjectures in birational geometry, people have been studying the birational geometry of foliations for a long time. The birational geometry of foliations is rich, but also differs strikingly in many aspects from the birational geometry of varieties. On one hand, foliations, as a powerful tool, can unveil the properties of varieties themselves surprisingly and deeply. For example, foliations play a crucial role in the proof of several key cases of the abundance conjecture for threefolds and the characterization of uniruled compact Kähler manifolds. On the other hand, many essential difficulties have emerged during the study of foliations. For example, abundance conjecture, effective birationality, Bertini-type theorems and many more important results in the birational geometry of varieties fail for foliations, and there is no hope to remedy this situation for just foliations. Two extremely essential and important divergences are the failure of finite generation and the failure of boundedness for foliations.

I will discuss some joint work with Paolo Cascini, Jingjun Han, Jihao Liu, Calum Spicer, Roberto Svaldi and Lingyao Xie where the notion of adjoint foliated structures is studied to overcome the aforementioned pathological behavior and difficulties of the birational geometry of foliations, and this methodology works. In particular, we prove the finite generation of the canonical rings and boundedness results for algebraically integrable adjoint foliated structures which can be viewed as analogues of the classical results for varieties.