The Couette-Taylor problem, a flow between two concentric rotating cylinders, has been widely studied as a good subject of the study of pattern fomation and transition to turbulence. Consider the case where the inner cylinder is rotating with uniform speed and the outer one is at rest. If the rotating speed is sufficiently small, a laminar flow (Couette flow) is stable. When the rotating speed increases, beyond a certain value of the rotating speed, a vortex flow pattern (Taylor vortex) appears. For viscous incompressible fluids, the occurrence of the Taylor vortex was shown to solve a bifurcation problem for the incompressible Navier-Stokes equations. In this talk, this problem will be considered for viscous compressible fluids. The spectrum of the linearized operator around the Couette flow is investigated and the bifurcation of the compressible Taylor vortex is proved when the Mach number is sufficiently small. It is also proved that the compressible Taylor vortex converges to the incompressible one when the Mach number tends to zero. This talk is based on a joint work with Prof. Takaaki Nishida (Kyoto University) and Ms. Yuka Teramoto (Kyushu University).