In this talk, I will present a fracton gravity gauge theory based on a covariant version of the monopole-dipole-momentum algebra. In the presence of a suitable background field, the theory includes a massive two-form and a massless symmetric two-index tensor. The latter transforms only under longitudinal diffeomorphisms, making the massless sector similar to linearized gravity, but with additional modes of lower spin. I will show that the theory can be consistently coupled to a curved background metric and discuss its possible interaction terms with itself and with matter. In addition, I will present a map between solutions of linearized gravity in Kerr-Schild form and solutions of fracton gravity coupled to matter.