  This survey is an introduction to the classification of Fatou components in
holomorphic dynamics. We start with the description of the Fatou and Julia sets
for rational maps of the Riemann sphere, and finish with an account of the
recent results on Fatou components for polynomial skew-products in complex
dimension two, where we focus on the key steps in the construction giving the
existence of a wandering domain for a polynomial endomorphism of
$\mathbb{C}^2$.
