Abstract: Fix an irrational number $\alpha$ and a smooth, positive, real function $p$ on the circle. If current position is $x$ then in the next step jump to $x+\alpha$ with probability $p(x)$ or to $x-\alpha$ with probability $1-p(x)$. Throughout the talk I will recall results of Sinai and Conze-Guivarc'h concerning uniqueness of stationary distribution for this random walk and discuss its mixing properties.