Let f be a holomorphic selfmap of a complex manifold X. A point z in X is said to lie in the Fatou set if the family of iterates is a normal family in a neighborhood of z. A connected component of the Fatou set is called a Fatou component. As all the orbits of a Fatou component behave similarly, understanding Fatou components is an important step in understanding complex dynamical systems. For rational functions in the Riemann sphere Fatou components are quite well understood: every Fatou component is preperiodic and periodic Fatou components have been completely classified. Neither is true in higher dimensions, although there has been some progress towards the description of periodic Fatou components. I will review what is known in the literature and present some recent results. The talk will be geared towards complex analysts, not dynamicists. This is joint work with Mikhael Lyubich.