We derive an exact theory of three-dimensional steady separation and reattachment using nonlinear dynamical systems methods. Specifically, we obtain criteria for separation points and separation lines on fixed no-slip boundaries in compressible flows. These criteria imply that there are only four basic separation patterns with well-defined separation surfaces. We also derive a first-order prediction for the separation surface using wall-based quantities; we verify this prediction using flow models obtained from local expansions of the Navier–Stokes equations.