Author’s abstract: A space curve in a Euclidean 3-space \(\mathbb E^3\) is called a rectifying curve if its position vector field always lies in its rectifying plane. The notion of rectifying curves was introduced by the author in [Am. Math. Mon. 110, No. 2, 147–152 (2003; Zbl 1035.53003)]. In this present article, we introduce and study the notion of rectifying submanifolds in Euclidean spaces. In particular, we prove that a Euclidean submanifold is rectifying if and only if the tangential component of its position vector field is a concurrent vector field. Moreover, rectifying submanifolds with arbitrary codimension are completely determined.