Abstract: The celebrated reduction theory of formal connections is due to Hukuhara, Levelt, Turrittin, and Babbitt-Varadarajan, among others. In this talk, we will demonstrate the parahoric reduction theory of formal parahoric connections, which generalizes the aforementioned results and also extends Boalch’s result for the case of regular singularities. As applications, we will establish the equivalence between extrinsic and intrinsic definitions of regular singularities, as well as a parahoric version of Frenkel-Zhu’s Borel reduction theorem for formal connections. Some further questions will be proposed as well. This is based on a recent joint work with Z. Hu, R. Sun, and R. Zong.