Let R be a (commutative integral) domain with quotient field K, let F be a field extension of K, and let \(R^{'}_F\) be the integral closure of R in F. It is proved that a quasi-local domain (R, M) is a going-down domain if and only if, for each nonzero nonmaximal prime ideal P of R, \(R^{'}_F\) is the intersection of the set of valuation rings of F that contain R, are centered on M, and have a prime ideal \(\mathcal {P}\) such that \(\mathcal {P} \cap R=P\) . Consequences include some new characterizations of arbitrary (that is, not necessarily quasi-local) going-down domains and, for any such domain R, some descriptions of \(R^{'}_F\) as an intersection of certain sets of valuation rings of F that contain R. Within a universe of domains with only finitely many valuation overrings, such descriptions of \(R^{'}_F\) can be conjoined with a property that is intermediate between “treed domain” and “going-down domain” to furnish several new characterizations of going-down domains.