In set theory and related branches of mathematics, a collection ${\displaystyle F}$ of subsets of a given set ${\displaystyle S}$ is called a family of subsets of ${\displaystyle S}$, or a family of sets over ${\displaystyle S.}$ More generally, a collection of any sets whatsoever is called a family of sets, set family, or a set system.

The term "collection" is used here because, in some contexts, a family of sets may be allowed to contain repeated copies of any given member, and in other contexts it may form a proper class rather than a set.

A finite family of subsets of a finite set ${\displaystyle S}$ is also called a hypergraph. The subject of extremal set theory concerns the largest and smallest examples of families of sets satisfying certain restrictions.