In this talk we consider the two-dimensional stationary Navier-Stokes equations describing the viscous incompressible flows around a rotating obstacle. We will show the existence and the asymptotic behavior at spatial infinity of stationary solutions when the speed of rotation is sufficiently small. The stability of these stationary solutions is still widely open, and we prove the local L^2 stability when the obstacle is a unit disk. A part of this talk is based on the joint work with Mitsuo Higaki (Tohoku university) and Yuu Nakahara (Tohoku university).