A vertex-based finite volume method for solving the three-dimensional compressible Reynolds-averaged Navier-Stokes equations is presented for calculating turbomachinery cascade flows. A discretization scheme for the viscous terms is proposed. The Baldwin-Lomax algebraic turbulence model is used. The scheme is verified against laminar and turbulent flows over a flat plate. Two- and three-dimensional computations were carried out for a low-pressure turbine cascade at design and off-design conditions. Results are compared with inviscid solutions and experimental data.