The remarkable Non-wandering domain theorem due to Sullivan leads to a complete classification of the dynamics for a rational function on its Fatou set. Up to now, the generalization of Sulllivan's theorem in high dimension focus on polynomial skew products. In the case we essentially need to study the semi-local theory, i.e. to study the Fatou set of polynomial skew products in a neighborhood of an invariant fiber which is attracting, parabolic or elliptic. In this talk I will overview the previous results on all these three kinds of polynomial skew products, and present a new theorem on the attracting case. The theorem states that there are no wandering domains for a polynomial skew product with an attracting invariant fiber when the multiplier is small.