The work, addressing subsonic, supersonic and hypersonic boundary-layer instability, is motivated by the need for more understanding of compressible transition at high global Reynolds numbers Re. In the supersonic case, the so-called 'first modes ' of instability found/suggested by previous Orr-Sommerfeld computations can be identified as triple-deck oblique ones, directed outside the local wave-Machcone directions. Less oblique instability modes inside are not of Orr-Sommerfeld form since they are substantially affected by non-parallel flow effects. The maximum linear growth rates are determined for a range of supersonic and subsonic free-stream Mach numbers M,, and comparisons are made with previous computations, showing fairly good agreement at moderate Mach numbers. A second mound of unstable wavenumbers and frequencies is also evident. I n addition, the nonlinear version is set up and emphasized (with attention drawn to a recent paper by the author (1988a) showing the possibility of nonlinear break-up), and certain extremes are examined including those of transonic and hypersonic boundary layers. In the hypersonic limit a new regime is found (for many conditions, including the insulated plate), namely M , -Re&, in which non-parallel-flow effects enter to control the main disturbances, and it is concluded that the restriction M , 4 Re& applies to the Orr-Sommerfeld approach. This is a very severe restriction in practice.