Developing advanced magnetic divertor configurations to address the coupling of heat and particle exhaust with impurity control is one of the major challenges currently constraining the further development of fusion research, and therefore has become the focus of extensive attention in recent years. In J-TEXT, several new divertor configurations, including the high-field-side single-null poloidal divertor and the island divertor, as well as their associated fundamental edge divertor plasma physics have recently been investigated. The purpose of this paper is to briefly summarize the latest progress and achievements in this relevant research field on J-TEXT in the past few years.