Magnetorheological (MR) fluids are suspensions of small particles whose apparent rheological properties can be altered dramatically by applying a magnetic field. For example, magnetic flux densities of the order of 1 Tesla can induce a yield stress of the order of 100 kPa in an otherwise essentially Newtonian fluid. After a brief introduction to magnetorheology, including a few of the more common applications, four vignettes of experimental observations and resulting modeling challenges will be presented. In the first vignette, transients in shear flow rheology observed for large applied magnetic field strengths are addressed. These transients are associated with the formation of lamellae within the suspension, whose dynamics can be modeled at the particulate or continuum levels. In the second vignette, unexpectedly large yield stresses observed for suspensions with bidisperse particle size distributions are described. Particle-level modeling reveals the mechanism, but predicting the magnitude of the enhancement remains a challenge. The third vignette examines effects of friction, which only appear at large concentrations. Observations are similar to jamming transitions observed in similar systems. The last vignette examines a surprising enhancement caused by replacing magnetizable particles with nonmagnetizable particles in MR fluids.