In this article, we study symmetric ( v, k ,λ ) designs admitting a flag-transitive and point- primitive automorphism group G whose socle is PSU 4 (q). We prove that there exist eight non-isomorphic such designs for which λ ∈ { 3 , 6 , 18 } and G is either PSU 4 (2), or PSU 4 (2) : 2.
