In the past five years, working largely independently, five groups of researchers have proposed low-dimensional models of the behavior of parallel shear flows at high Reynolds numbers. These models are compared, and it is found that they are more similar than their authors have recognized. Among other similarities, most of them exhibit a threshold amplitude&egr;=O(R&agr;)asR→∞for some &agr;<−1, whereRis the Reynolds number, for perturbations of the laminar state that may excite transition to turbulence. The reason for this behavior in each case is an interaction of non-normal linear effects with quadratic nonlinearities. ©1997 American Institute of Physics.