It has long been recognized that the fluid mechanics of gas-phase microflows can differ significantly from the macroscopic world. Non-equilibrium effects such as rarefaction and gas-surface interactions need to be taken into account, and it is well known that the no-slip boundary condition of the Navier-Stokes equations is no longer valid. Following ideas proposed by Maxwell, it is generally accepted that the Navier-Stokes equations can be extended into the slip-flow regime, provided the Knudsen number is less than 10-1. Improvements in micro-fabrication techniques, however, are now enabling devices to be constructed with sub-micron feature sizes. At this scale, the flow will depart even further from equilibrium and will enter the transition regime. In recent years, there has been considerable success in the implementation of second-order slip-boundary conditions to extend the Navier-Stokes equations into the transition regime. Unfortunately, as yet, no consensus has been reached on the correct form of higher-order approach, with theoretical and experimental studies revealing large discrepancies in the magnitude of the second-order slip coefficient. It is believed that these discrepancies can be explained by the fact that continuum flow analyses neglect the Knudsen layer, which extends approximately one mean-free path from the channel wall. In addition, comparisons between kinetic and continuum slip-boundary formulations reveal another important source of error due to different definitions in the first-order slip coefficient. The paper explains how these discrepancies have arisen and describes future research directions that may help reconcile the different forms of higher-order approach.