The ordered weighted average (OWA) operator has been widely used in decision-making. In many situations, however, providing crisp numbers for either the sub-criteria or the weights is problematic (there could be uncertainties about them), and it is more meaningful to provide intervals, type-1 fuzzy sets (T1 FSs), interval type-2 fuzzy sets (IT2 FSs), or a mixture of all of these, for the sub-criteria and weights. Two fuzzy extensions of the OWA, ordered fuzzy weighted averages for T1 FSs and ordered linguistic weighted averages for IT2 FSs, as wells as procedures for computing them, are introduced in this paper. They are compared with Zhou et al.'s T1 and IT2 fuzzy extensions of the OWA. Examples show that our extensions may give different results from Zhou et al.'s extensions when the legs of the FSs have intersections. Because our extensions coincide with the intuition of “FS in its entirety,” they are the suggested ones to use.