Superposition operators on spaces of analytic functions

If $\varphi $ is an entire function then the superposition operator $S_\varphi $ is defined by
$$S_\varphi (f)(z)=\varphi \left (f(z)\right ),\quad f\in Hol (\mathbb D),\quad z\in \mathbb D.$$
If $X$ and $Y$ are two subspaces of $ Hol(\mathbb D)$, the question is: For which entire functions $\varphi$ does the operator $S_\varphi$ map $X$ into $Y$?, and when is continuous and/or bounded?
This question has been solved for a good number of pairs $(X,\,Y)$ of spaces of analytic functions in the disc by distint authors including Cámera, Giménez, Buckley, Fernández, Bonet, Vukotic and others.
After presenting some of these results, I will turn to deal with a number of contributions to the topic I have obtain in collaboration with some colleagues in the last few years.