A large class of $4d$ $\mathcal{N}=2$ superconformal field theories arise as compactifications of a $6d$ $(2,0)$ theory of type $\mathfrak{j}=A,D,E$ on a punctured Riemann surface, $C$. These theories can be classified by listing the allowed fixtures and cylinders which can occur in a pants decomposition of $C$, and giving the rules for gluing them together. Different pants decompositions of the same surface give different weakly-coupled presentations of the same underlying SCFT, related by S-duality. An even larger class of theories can be constructed in this way by including ``twisted" punctures, which carry a non-trivial action of the outer-automorphism group of $\mathfrak{j}$. In this dissertation, we discuss the classification procedure for twisted theories of type $D_N$, as well as for twisted and untwisted theories of type $E_6$. Using these results, we write the Seiberg-Witten solutions for all $Spin(n)$ gauge theories with matter in spinor representations which can be realized by compactifying the $(2,0)$ theory. We also study a family of SCFTs arising from the twisted $A_{2N}$ series, whose twisted punctures are still not fully-understood.