The stability characteristics of some fluid flows at high Taylor or Gortler numbers are determined using perturbation methods. In particular, the stability characteristics of some fully developed flows between concentric cylinders driven either by a pressure gradient or the motion of the inner cylinder are investigated. The asymptotic structure of short-wavelength disturbances to these flows is obtained and used as a basis for a formal perturbation solution to the corresponding stability problem appropriate to a developing boundary layer. The non-parallel effect of the basic flow on the condition for neutral stability is discussed. The results obtained suggest that the disturbances are concentrated in internal viscous or critical layers well away from the wall and the free stream. The stability of a boundary layer on a concave wall to Görtler vortices that propagate downstream is also considered. These modes are found to be more stable than the usual time-independent modes and they propagate downstream with the speed of the basic flow in the critical layer. Some comparison with previous experimental and theoretical work is given.