  It is known, from results of B. MacCluer and J. Shapiro (1986), that every
composition operator which is compact on the Hardy space $H^p$, $1 \leq p <
\infty$, is also compact on the Bergman space ${\mathfrak B}^p = L^p_a (\D)$.
In this survey, after having described the above known results, we consider
Hardy-Orlicz $H^\Psi$ and Bergman-Orlicz ${\mathfrak B}^\Psi$ spaces,
characterize the compactness of their composition operators, and show that
there exist Orlicz functions for which there are composition operators which
are compact on $H^\Psi$ but not on ${\mathfrak B}^\Psi$.
