An analysis is given for the viscous hypersonic flow past slender sharp wedges and unyawed cones. The leading edge or nose region treated encompasses the merged layer regime in which the shock thickness is not small in comparison with the viscous layer thickness but for which in the axisymmetric case the shock thickness is small in comparison with the radius of transverse shock curvature. A Navier-Stokes flow is assumed, and both the shock structure and viscous layer are taken to be locally-similar of boundary layer type. The analysis follows that used by Shorenstein and Probstein for the corresponding flat plat problem. For axisymmetric cone flow, a modification of a transformation introduced by Probstein and Elliott is given to reduce the axisymmetric viscous layer equations including transverse curvature to 'exactly' two-dimensional form. Results for the state conditions behind the shock, and the wall pressure, skin friction, and heat transfer rate are found to be in agreement with available experimental data.