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General $\mathcal{N}=(1,0)$ supergravity-matter systems in six dimensions maybe described using one of the two fully fledged superspace formulations forconformal supergravity: (i) $\mathsf{SU}(2)$ superspace; and (ii) conformalsuperspace. With motivation to develop rigid supersymmetric field theories incurved space, this paper is devoted to the study of the geometric symmetries ofsupergravity backgrounds. In particular, we introduce the notion of a conformalKilling spinor superfield $\epsilon^\alpha$, which proves to generate extendedsuperconformal transformations. Among its cousins are the conformal Killingvector $\xi^a$ and tensor $\zeta^{a(n)}$ superfields. The former parametriseconformal isometries of supergravity backgrounds, which in turn yieldsymmetries of every superconformal field theory. Meanwhile, the conformalKilling tensors of a given background are associated with higher symmetries ofthe hypermultiplet. By studying the higher symmetries of a non-conformal vectormultiplet we introduce the concept of a Killing tensor superfield. We alsoanalyse the problem of computing higher symmetries for the conformald'Alembertian in curved space and demonstrate that, beyond the first-ordercase, these operators are defined only on conformally flat backgrounds.

Symmetries of $\mathcal{N} = (1,0)$ supergravity backgrounds in six dimensions