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A bstract We carry out a systematic study of 4d    $$ \mathcal{N} $$  N = 2 preserving S-folds of F-theory 7-branes and the worldvolume theories on D3-branes probing them. They consist of two infinite series of theories, which we denote following [1, 2] by    $$ {\mathcal{S}}_{G,\mathrm{\ell}}^{(r)} $$  S G , ℓ r for  ℓ  = 2 ,  3 ,  4 and    $$ {\mathcal{T}}_{G,\mathrm{\ell}}^{(r)} $$  T G , ℓ r for  ℓ  = 2 ,  3 ,  4 ,  5 ,  6. Their distinction lies in the discrete torsion carried by the S-fold and in the difference in the asymptotic holonomy of the gauge bundle on the 7-brane. We study various properties of these theories, using diverse field theoretical and string theoretical methods.

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