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We obtain exhaustive results and treat in a unified way the question ofboundedness, compactness, and weak compactness of composition operators fromthe Bloch space into any space from a large family of conformally invariantspaces that includes the classical spaces like $BMOA$, $Q_\alpha$, and analyticBesov spaces $B^p$. In particular, by combining techniques from both complexand functional analysis, we prove that in this setting weak compactness isequivalent to compactness. For the operators into the corresponding "small"spaces we also characterize the boundedness and show that it is equivalent tocompactness.

Compact and weakly compact composition operators from the Bloch space into M\"obius invariant spaces