Motivated by recent experimental studies of rheological hysteresis in soft glassy materials, we study numerically strain rate sweeps in simple yield stress fluids and viscosity bifurcating yield stress fluids. Our simulations of downward followed by upward strain rate sweeps, performed within fluidity models and the soft glassy rheology model, successfully capture the experimentally observed monotonic decrease of the area of the rheological hysteresis loop with sweep time in simple yield stress fluids, and the bell shaped dependence of hysteresis loop area on sweep time in viscosity bifurcating fluids. We provide arguments explaining these two different functional forms in terms of differing tendencies of simple and viscosity bifurcating fluids to form shear-bands during the sweeps, and show that the banding behaviour captured by our simulations indeed agrees with that reported experimentally. We also discuss the difference in hysteresis behaviour between inelastic and viscoelastic fluids. Our simulations qualitatively agree with the experimental data discussed here for four different soft glassy materials.Many soft materials, including emulsions, 1 foams, 2 colloids, 3,4 microgels 5 and star polymers 6 display exotic rheological behaviour intermediate between that of liquids and solids.7,8 At rest, or under low imposed strains, they behave as weak elastic solids and often also show ageing behaviour, in which a sample becomes progressively more solid-like as a function of the time since it was prepared.9-11 In contrast, for imposed stresses larger than a threshold yield stress σ y , they show more liquid-like response, although the resulting flow might be spatially homogeneous or heterogeneous, 12 steady or strongly time dependent, 13 depending on the nature of the interactions between the constituent particles, 1,13-15 the shear history [16][17][18] and even the boundary conditions. 19,20 Such phenomena have been attributed to the generic presence in these 'soft glassy materials' (SGMs) 21,22 of structural disorder and metastability.Understanding the rheology of soft glasses remains the focus of considerable ongoing debate.23,24 Many practical applications require the determination of the flow curve σ(γ), which links the shear stress σ to the shear rateγ under conditions of a steady shear flow. Experimentally, the measurement of this curve usually involves sweeping the shear rate (or shear stress) up or down over some prescribed interval, with some prescribed temporal duration for the sweep. While the aim is to measure the steady-state flow curve, in practice a time-dependent response is often seen. In any fluid with a fixed intrinsic relaxation timescale, the departure from steady-state can be quantified by comparing that relaxation timescale to the time of the sweep. In a glassy material, however, the typical relaxation timescale is a complicated function of the imposed flow history. In practice, therefore, the usual strategy is to compare flow curve measurements obtained in a sequence comprising an upward fo...