Abstract: If one searches for k-dimensional submanifolds with critical k-dimensional volume in a Riemannian manifold, then one is led towards non-linear elliptic partial differential equations involving the mean curvature vector of the submanifold. I will present new constructions of volume-critical submanifolds of the sphere in two contexts: hypersurfaces with constant mean curvature in spheres of any dimension; and Legendrian submanifolds in spheres of odd dimension that are stationary under variations preserving the contact structure. These are constructed by solving the associated non-linear elliptic PDE using singular perturbation theory in its geometric formulation known as "gluing". I will then highlight some of the analytic and geometric similarities between these two contexts.