This talk considers immiscible viscous multilayer flows driven by gravity, pressure gradients or the motion of one of the confining boundaries. Such flows can become unstable at arbitrarily small (and even zero) Reynolds numbers, even when analogous single-phase flows are stable (e.g. Couette flows). As a result applications are found in microfluidic geometries and other micro scale flows. Of particular interest are flows driven out of equilibrium by external effects such as imposed electric field or blowing and suction through a boundary. We will present asymptotically derived models motivated by physical experiments, that are amenable to detailed analysis and extensive computations. Direct numerical simulations will also be presented and some novel electric field-induced dynamics will be described that could be useful in mixing and pumping on the micro scale.