  Let $\Omega_1,\Omega_2\subset {\mathbb C}$ be bounded domains. Let
$\phi:\Omega_1\rightarrow \Omega_2$ holomorphic in $\Omega_1$ and belonging to
$W^{1,\infty}_{\Omega_2}(\Omega_1)$. We study the composition operators
$f\mapsto f\circ\phi$ on generalized Hardy spaces on $\Omega_2$, recently
considered in \cite{bfl, BLRR}. In particular, we provide necessary and/or
sufficient conditions on $\phi$, depending on the geometry of the domains,
ensuring that these operators are bounded, invertible, isometric or compact.
Some of our results are new even for Hardy spaces of analytic functions.
