Let denote the unit disc in the complex plane and let denote the set of all functions holomorphic on . For a point , let
The spaces BMOA and the Bloch space are defined by
Then and . Let such that , and, for , let . Let denote either of the spaces BMOA or . The authors prove that the composition operator is a compact operator on the space if and only if . For the space BMOA, this improves a result of the first author [Sci. China, Ser. A 50, No. 7, 997–1004 (2007; Zbl 1126.30023)].