Wet-steam flows through 2-D and 3-D multiple-stage turbine cascades are numerically investigated. Fundamental equations contain conservation laws of water vapor, water liquid, the number density of nucleus, momentum equations, and the energy equation. The SST turbulence model is employed to predict the eddy viscosity. The classical condensation theory is used for modeling homogeneous and heterogeneous condensations. These equations are solved by the fourth-order compact MUSCL TVD scheme and LU-SGS scheme. A parallelized LU-SGS method based on the pipelining method is newly employed for the parallel-implicit time-integration. As numerical examples, 2-D and 3-D flows of wet steam through two-stage stator-rotor cascade channels in a steam turbine are computed.