  In this article, we study flag-transitive automorphism groups of non-trivial
symmetric $(v, k, \lambda)$ designs, where $\lambda$ divides $k$ and $k\geq
\lambda^2$. We show that such an automorphism group is either point-primitive
of affine or almost simple type, or point-imprimitive with parameters
$v=\lambda^{2}(\lambda+2)$ and $k=\lambda(\lambda+1)$, for some positive
integer $\lambda$. We also provide some examples in both possibilities.
