The dynamics inside periodic components of the stable set has a strong link with classical theorems of complex analysis like the Denjoy-Wolff Theorem about analytic maps of the unit disk. The fractal boundaries of such components arising so naturally from iteration often present interesting topological properties which may play a role when trying to transfer results from the unit disk back to the dynamical plane. However, if the components are not periodic but wandering, we need to reach further and consider non-autonomous iteration. Starting from periodic components, I aim to present some recent results about the dynamics inside wandering domains and also on their boundaries. Many of the results are proven in the very general setting of non-autonomous dynamics or even for sequences of holomorphic maps.