  We compute the Nash blow-up of a cominuscule Schubert variety. In particular,
we show that the Nash blow-up is algebraically isomorphic to another Schubert
variety of the same Lie type. As a consequence, we give a new characterization
of the smooth locus and, for Grassmannian Schubert varieties, determine when
the Nash blow-up is a resolution of singularities. We also study the induced
torus action on the Nash blow-up and give a bijection between its torus fixed
points and Peterson translates on the Schubert variety.
