Summary: The generalized ordered weighted averaging (GOWA) operators are a new class of operators, which were introduced by \textit{R. R. Yager} [Fuzzy Optim. Decis. Mak. 3, No.~1, 93--107 (2004; Zbl 1057.90032)]. However, it seems that there is no investigation on these aggregation operators to deal with intuitionistic fuzzy or interval-valued intuitionistic fuzzy information. In this paper, we first develop some new generalized aggregation operators, such as generalized intuitionistic fuzzy weighted averaging operator, generalized intuitionistic fuzzy ordered weighted averaging operator, generalized intuitionistic fuzzy hybrid averaging operator, generalized interval-valued intuitionistic fuzzy weighted averaging operator, generalized interval-valued intuitionistic fuzzy ordered weighted averaging operator, generalized interval-valued intuitionistic fuzzy hybrid average operator, which extend the GOWA operators to accommodate the environment in which the given arguments are both intuitionistic fuzzy sets that are characterized by a membership function and a nonmembership function, and interval-valued intuitionistic fuzzy sets, whose fundamental characteristic is that the values of its membership function and nonmembership function are intervals rather than exact numbers, and study their properties. Then, we apply them to multiple attribute decision making with intuitionistic fuzzy or interval-valued intuitionistic fuzzy information.