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