An accurate and rapid method is presented for solution of the general equations of compressible, steady, laminar-boundary-layer flow. The method allows arbitrary conditions on all of the following: pressure gradient, surface temperature and its gradient, heat transfer, mass transfer, and fluid properties. Also, the method can calculate the second-order effect of transverse curvature. The only restrictions on the method are that the body be axially sym metric or two-dimensional and that no dissociation of the fluid occurs. The equations that solved are developed from the Navier-Stokes and energy equations by an order-of-magnitude analysis. They differ from the conventional boundary-layer equations of Prandtl only in that the second-order terms that include transverse curvature are retained. The method of solution consists of replacing the partial derivatives with respect to the flow direction by finite differences, while retaining the derivatives in a direction normal to the boundary, so that the partial differential equations become approximated by ordinary differential equations.