Broué conjecture is one of the most important conjectures in the representation theory of finite groups. Broué conjecture over algebraically closed fields is due to M. Broué. Chuang and Rouquier proved that the conjecture (actually over arbitrary fields) holds for symmetric groups and general linear groups (2008). By the work of Chuang and Rouquier, it was observed that the conjecture might hold for arbitrary fields. Craven and Rouquier investigated this observation (2013) and they proved that Broué conjecture over arbitrary fields holds for principal blocks for finite groups with abelian Sylow 2-subgroups and with Sylow 3-group of order 9. Inspired by Galois-Alperin-McKay conjecture due to Navarro, Kessar and Linckelmann formally raised Broué conjecture over arbitrary fields (2018) and proved that Broué conjecture over arbitrary fields holds for cyclic blocks. However, all these investigations above for Broué conjecture over arbitrary fields are case by case. In this talk, we aim to generally study Broué conjecture from over algebraically closed fields to over arbitrary fields. As a consequence of this study, we prove that Puig’s finiteness conjecture for inertial blocks is true.