A brief review is given on the stability of two-fluid systems. Our interest is primarily driven by drag reduction using superhydrophobic surfaces (SHS) or liquid-infused surfaces (LIS) where the longevity and performance strongly depends on the flow stability. Although the review is limited to immiscible, incompressible, Newtonian fluids with constant properties, the subject is rich in complexity. We focus on three canonical plane parallel flows as part of the general problem: pressure-driven flow, shear-driven flow, and flow down an inclined plane. Based on the linear stability, the flow may become unstable to three modes of instabilities: a Tollmein–Schlichting wave in either the upper fluid layer or the lower fluid layer, and an interfacial mode. These instabilities may be further categorized according to the physical mechanisms that drive them. Particular aspects of weakly nonlinear analyses are also discussed, and some directions for future research are suggested.