A numerical method to solve the three-dimensional Navier-Stokes equations for the flow in transonic turbine stages with tip gap leakage is presented. Viscous flow in a transonic turbine stage has been simulated. The high pressure difference at the rotor blade tip results in a supersonic jet. The relative motion of the casing wall is oriented against the tip leakage flow and tends to reduce it. Very large velocity gradients in the tip region pose a challenge for the numerical simulation. Computational results are compared with experimental data obtained in operation. Measurements include data for the tip leakage jet. The numerical method is based on a conservative finite volume cell–vertex scheme in cylindrical coordinates with central difference approximation and Runge–Kutta time stepping. Convergence is accelerated by use of a multigrid method and implicit residual smoothing with variable coefficients. The Baldwin–Lomax turbulence model is used for closure. The boundary condition treatment at inlet and outlet as well as the coupling of stator and rotor flow is achieved by use of non–reflective boundary conditions. The tip region is discretized by an additional grid within a multi-block approach.