We construct ergodic actions of compact quantum groups on C∗-algebras and von Neumann algebras, and exhibit phenomena of such actions that are of different nature from ergodic actions of compact groups. In particular, we construct: (1). an ergodic action of the compact quantum Au(Q) on the type IIIλ Powers factor Rλ for an appropriate positive Q ∈ GL(2, R); (2). an ergodic action of the compact quantum group Au(n) on the hyperfinite II1 factor R; (3). an ergodic action of the compact quantum group Au(Q) on the Cuntz algebra On for each positive matrix Q ∈ GL(n, C); (4). ergodic actions of compact quantum groups on the their homogeneous spaces, and an example of a non-homogeneous classical space that admits an ergodic action of a compact quantum group.