Recently, the author defined and classified rectifying submanifolds in Euclidean spaces in ; extending his earlier work on rectifying curves in Euclidean 3-space done in . In this article, first the author introduces the notion of rectifying submanifolds in an arbitrary Riemannian manifold. Then he defines torqued vector fields on Riemannian manifolds and classifies Riemannian manifolds which admit a torqued vector field. Finally, he characterizes and studies rectifying submanifolds in a Riemannian manifold equipped with a torqued vector field. Some related results and applications are also presented.