In this article, we study symmetric (v,k,λ) designs D with λ10 admitting flag-transitive and point-primitive almost simple automorphism group G with socle X. We, in particular, determine all such possible designs and show that such a design is isomorphic to a Desarguesian plane with parameter set (q2 + q + 1, q + 1, 1) and X = PSL3 (q), or D is isomorphic to one of the seventeen well-known designs.
