Ellis & Gottschalk introduced the regionally proximal relation in 1960 whereas the higher order regionally proximal relations for abelian group actions were introduced by Host, Kra & Maass in 2010. The interest in these relations lies in their role in the structural theory of topological dynamical systems. For amenable minimal group actions the regionally proximal relation is an equivalence relation but for some non-amenable minimal actions it is not. When it is an equivalence relation, the quotient by the regionally proximal relation is the maximal equicontinuous factor. In this talk I will introduce a generalization of the higher order regionally proximal relations suitable for an arbitrary acting group. The surprising new main result is that these generalized relations are always equivalence relations for an arbitrary minimally acting group. Moreover the generalized regionally proximal equivalence relation of order one corresponds to the maximal (compact) abelian group factor, yielding for the first time an explicit description of this factor in the category of non-abelian minimal actions. Joint work with Eli Glasner and XiangDong Ye.