Abstract: Partial masslessness occurs when an additional scalar gauge symmetry is present in a massive gravity theory, reducing the number of propagating degrees of freedom from five to four. For example, this is known to happen at the Higuchi bound for a massive graviton propagating on Einstein space-times. But the existence of a fully non-linear theory whose graviton is partially massless remains an open question. In this talk, I will start by an introduction on massive and partially massless gravitons on curved background spacetimes. Then, starting from a recently constructed theory for a massive graviton that propagates the correct number of degrees of freedom on an arbitrary spacetime, I will show that that a partially massless graviton can propagate on a large set of spacetimes which are not Einstein spacetimes.