In this talk I will present a joint work with W. He and Y. Jiao. We study the random walks and group actions by circle diffeomorphisms. Under mild assumptions we establish several results on the dimensional properties of invariant measures and attractors (limit sets or minimal sets). Our results include exact dimensionality and dimension formula of stationary measures, variational principles for dimensions in various settings, estimates of Hausdorff dimension of exceptional minimal set, etc. We also show an approximation theorem for random walks by circle diffeomorphisms which is analogous to the results of Katok, Avila-Crovisier-Wilkinson, Morris-Shmerkin.