A submanifold L of the Euclidean space is called taut if for a generic point x the squared distance function to x is a perfect Morse function. This defnition can be generalized to a submanifolds of a Riemannian manifold by saying L is taut if the energy functional on a generic homotopy fiber of the inclusion is a perfect Morse function. We will consider a singular Riemannian foliation, which is a generalization of the partition given by the orbits of an isometric action, and assume that its leaves are taut. We will first survey known results that go back at least to the work of Bott and Samelson from the fifties and then explain a partial characterization of such foliations.