In this article, we determine all pairs (D, G), where D is a 2-design with gcd(r, λ) = 1 and G is a flag-transitive almost simple automorphism group of D whose socle is PSU(n, q) with (n,q) ̸= (3,2). We prove that such a design belongs to one of the two infinite families of Hermitian unitals and Witt-Bose-Shrikhande spaces, or it is isomorphic to a design with parameters (6,3,2), (7,3,1), (8,4,3), (10,6,5), (11,5,2) or (28,7,2). In particular, if D is symmetric, then it is either the Fano plane with parameters (7, 3, 1), or the unique Hadamard design with parameters (11, 5, 2).
