In this talk, we consider the asymptotic behavior of solutions to the Cauchy problem for the scalar viscous conservation law where the far field states are prescribed. Especially, we deal with the case when the flux function is a non-convex nonlinear function, and also the viscosity is a nonlinear function. When the corresponding Riemann problem for the hyperbolic part admits a Riemann solution which consists of single shock wave, it is proved that the solution of the Cauchy problem tends toward the viscous shock wave as time goes to infinity, provided the initial perturbation is suitably small.