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This paper determined all pairs $(\mathcal{D},G)$ where $\mathcal{D}$ is anon-symmetric 2-$(v,k,\lambda)$ design with $(r,\lambda)=1$ and $G$ is thealmost simple flag-transitive automorphism group of $\mathcal{D}$ with anexceptional socle of Lie type. We prove that if $T\trianglelefteq G\leq Aut(T)$where $T$ is an exceptional group of Lie type, then $T$ must be the Ree groupor Suzuki group, and there are five classes of non-isomorphic designs$\mathcal{D}$.

Flag-transitive non-symmetric $2$-designs with $(r,\lambda)=1$ and exceptional groups of Lie type