In this article, we study symmetric designs admitting a flag-transitive and point-primitive automorphism group whose socle is a symplectic group of dimension four. We prove a reduction theorem severely restricting the possible parameters of such designs, and we show that such a parameter set belongs to one of the eight infinite families of parameter sets. For four families of these parameter sets, we know two sporadic examples of symmetric designs with parameters (45, 12, 3) and (36, 15, 6) arose from PSp4 (3) and two infinite families of 2-designs namely the complement of a point-hyperplane design of PG(3,q) and the complement of a Higman design. The existence of any symmetric designs with the remaining four parameter sets is still open.
