The suppression of turbulence by mean flow shear is shown to apply to nonionized flows as well as plasmas. However, besides the criterion that the shearing rate exceed the turbulent decorrelation rate, there are three additional conditions. These stipulate that the shear flow must be stable, that turbulence must remain in the domain of flow shear for longer than an eddy turnover time, and that the dynamics should be two dimensional (2D). In nonionized flows, these conditions are not typically satisfied, explaining why shear suppression is not a familiar phenomenon in hydrodynamics. The three conditions are discussed in the context of nonionized and plasma flows. Two examples of suppression in nonionized flows are presented. One involves the formation of coherent structures in 2D Navier-Stokes turbulence and the other involves large-scale turbulence in the stratosphere. (C) 2000 American Institute of Physics. [S1070-664X(00)91805-6].