Let \(R\) be an integral domain, \(T\) an integral domain strictly containing \(R\). If \(R\) is not a field and there is no ring properly between \(R\) and \(T\), then \(T\) is named a minimal overring of \(R\). First of all the author shows that such a minimal overring, if it exists, is contained in the quotient field of \(R\). Then the paper is devoted to the investigation of the existence of minimal overrings for noetherian domains. In the one- dimensional case, minimal overrings are shown to exist always. For bigger dimensions, conditions are given such that minimal overrings exist or not. If the integral closure of \(R\) is finite over \(R\) and \(R\) has a maximal ideal of depth one, minimal overrings exist.
On the other hand, noetherian integrally closed domains such that all maximal ideals have depth greater or equal to 2 have no minimal overrings. The same is shown to be true for noetherian integral domains containing an infinite field and satisfying the same condition as above on the maximal ideals.