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In this article, we study symmetric $(v, k, \lambda)$ designs admitting aflag-transitive and point-primitive automorphism group $G$ whose socle is aprojective special unitary group of dimension at most five. We, in particular,determine all such possible parameters $(v, k, \lambda)$ and show that thereexist eight non-isomorphic of such designs for which $\lambda\in\{3,6,12, 16,18\}$ and $G$ is $PSU_{3}(3)$, $PSU_{3}(3):2$, $PSU_{4}(2)$ or $PSU_{4}(2):2$.

Symmetric designs and projective special unitary groups of dimension at most five