In this article, we study symmetric (v,k,λ) designs admitting a flag-transitive and point- primitive automorphism group G whose socle is isomorphic to a projective special linear group of dimension at most four. We, in particular, determine all such possible parame- ters (v,k,λ) and show that such a design belongs to one of two infinite families of point- hyperplane designs or it is isomorphic to a design with parameters (7, 3, 1), (7, 4, 2), (11, 5, 2), (11, 6, 3), (15, 8, 4), (35, 18, 9) or (56, 45, 18).
