Stability and bifurcation problem is considered for the compressible Navier-Stokes equations in a domain between two concentric cylinders. If the outer cylinder is at rest and the inner one rotates with sufficiently small angular velocity, a laminar flow, called the Couette flow, is stable. When the angular velocity of the inner cylinder increases, beyond a certain value of the angular velocity, the Couette flow becomes unstable and a vortex pattern, called the Taylor vortex, bifurcates and is observed stably. This phenomena is mathematically formulated as a bifurcation and stability problem. In a framework of the incompressible Navier-Stokes equations, the bifurcation and stability of the Taylor vortex has been studied in detail, while, for the compressible Navier-Stokes equations, only the bifurcation of the Taylor vortex has been known, but detailed structure of the bifurcation has remained unknown in many aspects. In this talk, the smoothness of the bifurcation curve and the stability of the compressible Taylor vortex will be shown under axisymmetric perturbations.