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Unsteady boundary-layer separation at high Reynolds numbers, Re, is considered on a theoretical and computational basis. Whenever an external inviscid flow induces a region of adverse pressure gradient near a wall, the development of recirculating eddies in the boundary layer is common. An unsteady viscous-inviscid interaction often follows in the form of a local boundary-layer eruption and abrupt ejection of near-wall vorticity into the external flow. The dynamics of this process, as it develops in an initially thin boundary layer, is considered. As interaction ensues, the flow focuses into a band which progressively narrows in the streamwise direction. The complex flow development is extremely difficult to resolve using conventional Eulerian methods; here the boundary-layer solutions are obtained using Lagrangian methods, wherein trajectories of a large number of fluid particles are computed. The algorithms developed are general but are applied here to the problem of the boundary-layer induced by a two-dimensional vortex above an infinite plane wall. Solutions are obtained for the limit problem Re approaches limit of infinity, and for Re large but finite using an interacting boundary-layer approach. The present results describe the initial stages of a strong unsteady viscous-inviscid interaction; apparently it is necessary to account for the effect of normal pressure variations to continue the interaction. Keywords: Vortices; Turbulent boundary layer; Turbulent bursts; Boundary-layer eruptions; Unsteady interactions; Boundary layer flow separation.