This review aims to steer a central course through the high Reynolds number theory of laminar flows, from its early classical beginnings and failures to its present-day status. The review is divided into three parts.

Part I concerns external flows. It deals in turn with the classical attached flow strategy and its deficiencies; the incompressible triple-deck structure; the compressible triple-deck, supersonic flow and breakaway separation; incompressible breakaway separation and applications; and, finally, other triple-deck situations and other external flow structures.

Part II concerns internal flows. Symmetric channel flows and axisymmetric pipe flows are considered first, followed by a discussion of non-symmetric channel flows and three-dimensional pipe flows.

Part III concerns recent work on special situations where the classical strategy is consistent with separation. This deals with: flow over an obstacle in a wall layer; marginal separation; and, finally, unsteady breakaway separation.

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