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**Shift Symmetries in (Anti) de Sitter Space**

A free massless scalar has an infinite number of shift symmetries in flat space. In (A)dS space, each of these symmetries is preserved only for particles with particular discrete masses. I will show how these shift symmetries generalize to massive higher-spin particles in (A)dS space and explain how these are related to partially massless symmetries. For the case of scalar fields, I discuss deformations of the symmetry algebras underlying these shift-symmetric theories and whether there exist invariant interactions. This leads to a ghost-free theory in (A)dS space that is invariant under a deformed quadratic shift symmetry, which reduces in flat space to the special Galileon. This theory has a rich structure of interactions that are completely fixed by the nonlinear symmetry, including a nontrivial potential. Lastly, I will speculate on possible generalizations to interacting massive higher-spin particles.