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