This talk will be in two parts. The first will describe some recent results
on the stability of multilayer shear flows of three or more immiscible viscous
fluids caused to flow by gravity and/or a pressure gradient. There are now at
least two free boundaries and asymptotic solutions will be described that
yield a system of coupled partial differential equations for the interfacial positions.
Some subtleties in the weakly nonlinear asymptotics will be pointed out.