The authors introduce and investigate a new class of monoids, called finitary monoids (here monoid means a commutative, cancellative semigroup with identity element). A monoid \(H\) (written multiplicatively) with group of invertible elements \(H^\times\) is called finitary if there is a finite subset \(U\subset H- H^\times\) such that \((H-H^\times)^n\subset UH\) for some positive integer \(n\) and \(\bigcap_{k\geq 0}a^kH=\emptyset\) for each \(a\in U\). The motivation for introducing finitary monoids comes from the theory of non-unique factorization in integral domains. For example, the multiplicative monoid \(R-\{0\}\) of a Noetherian integral domain \(R\) is finitary if and only if \(R\) is one-dimensional and semilocal. Other examples of finitary monoids include finitely generated monoids, \(v\)-Noetherian \(G\)-monoids, strongly primary monoids, and abstract congruence monoids.