We examine the effects of local turbulence Reynolds number R⋋ and inflow conditions on the magnitude of vorticity in plane turbulent wakes. Measurements of the spanwise component (ω₃) of the fluctuating vorticity vector ω = ω₁i + ω₂j + ω₃k (here the subscripts 1, 2 and 3 denote the streamwise, lateral and spanwise directions, respectively) are made in turbulent wakes of a screen and a circular cylinder. Lateral distributions of ω*₃ (normalized) in general depend on both R⋋ and inflow conditions. In the developing region, as the downstream distance x₁ increases, ω*₃ increases significantly in the screen wake but decreases slightly in the cylinder wake. Far downstream in the self-preserving region, ω*₃ increases linearly with R⋋ while it no longer varies with x₁ and depends weakly on the influence of inflow conditions. Our analysis suggests that these findings from the measurement of ω*₃ should apply for the magnitude of ω.