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In this article, we study $2$-designs with $\lambda=2$ admitting aflag-transitive almost simple automorphism group with socle a finite simpleexceptional group of Lie type, and we prove that such a $2$-design does notexist. In conclusion, we present a classification of $2$-designs with$\lambda=2$ admitting flag-transitive and point-primitive automorphism groupsof almost simple type, which states that such a $2$-design belongs to aninfinite family of $2$-designs with parameter set $((3^n-1)/2,3,2)$ and$X=PSL_n(3)$ for some $n\geq 3$, or it is isomorphic to the $2$-design withparameter set $(6,3,2)$, $(7,4,2)$, $(10,4,2)$, $(10,4,2)$, $(11,5,2)$,$(28,7,2)$, $(28,3,2)$, $(36,6,2)$, $(126,6,2)$ or $(176,8,2)$.

Almost simple groups as flag-transitive automorphism groups of 2-designs with {\lambda} = 2