We generalize the power averaging operators to interval-valued Atanassov’s intuitionistic fuzzy environments, and develop a series of generalized interval-valued Atanassov’s intuitionistic fuzzy power aggregation operators. The main advantages of these operators are that they not only accommodate situations in which the input arguments are interval-valued intuitionistic fuzzy numbers (IVIFNs), but also consider information about the relationship between the IVIFNs being fused. The properties of these operators are investigated and the relationships among these operators are discussed. Moreover, approaches to multiple attributes group decision making based on the proposed operators are given and two examples are illustrated to show the feasibility and validity of the new approaches to the application of multiple attributes group decision making.