AbstractCurrently no theory of the stably stratified atmospheric boundary layer (SBL) is generally accepted as definitive even for idealized cases. Nieuwstadt's theory, though a promising candidate, faces objections relating to upper boundary and steady‐state conditions, internal wave effects and the consistency of the model outside the strong‐stability limit. the aim of this paper is to examine the objections, improve the model, draw further deductions and compare with numerical models.We shall deduce from Nieuwstadt's model thatBo=RfcG2|f|/√3, whereBois the surface buoyancy‐flux,Gthe geostrophic wind speed,fthe Coriolis parameter andRfcthe critical value of the flux Richardson numberRf. Higher‐order expansion shows this value is an upper bound corresponding to the stable limitL/h→ 0. This is consistent with a similar bound onBoderived from independent energy arguments. By contrast, within present idealizations there is no bound on the surface cooling rate.Comparisons with a new series of large eddy simulations (LES) and other numerical models support the interpretation of Nieuwstadt's SBL as an idealized limiting case. the theory explains from first principles the observed sensitivity to small slopes. Interaction between sharp inversions and slopes may cause turbulence in thin layers. Coupling with the surface boundary conditions is a likely cause of intermittent turbulence, and explains features of Brost and Wyngaard's second‐order closure study. the formal singularity at the top of the SBL gives time‐scales for approach to inertial and heat equilibrium. Both, for separate reasons, are O(|f|−1).A natural extension to moderately stable and near‐neutral conditions agrees well with numerical model results, and provides a complete prediction of SBL structure from given surface heat flux and synoptic pressure gradient. the near‐neutral regime is narrow at high Rossby number. Comparison with LES supports both the local scaling approach and the gross predictions of the theory. the model gives insight into the limitations of Rossby‐similarity formulae. In particular, restrictions of domain sizehd(or similar background stability effects) may not become negligible even whenh/hd→ 0. Hence matching to neutral conditions is important in predicting even quite stable boundary layers. Wave effects, though non‐local in some respects, do not fundamentally change the Nieuwstadt picture for the mean structure. Retention in the model of the original value forRfcis recommended even if wave radiation perturbs localRf.In summary, the adapted Nieuwstadt theory seems to provide a definitive framework for the idealized SBL, from which other ‘perturbation’ effects may be assessed. the value ofBo/G2|f| gives a criterion for (a) quasi‐stead