In mathematics, a Fatou set is defined informally as the largest open set of points on which iterations of a given map (or collection of maps) have relatively tame long-term behaviour, in the sense that points that start close together stay close together. A map's Fatou set is the complement of its Julia set.

Less commonly, the term Fatou set or Fatou dust is sometimes used to refer to a Julia set that is not connected.

Fatou sets are named after French mathematician Pierre Fatou.