  In this article, we study symmetric $(v, k, \lambda)$ 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 $(v, k, \lambda)$ and show that there
exist 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 $PSU_{4}(2):2$.
