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S-folds are generalizations of orientifolds in type IIB string theory, suchthat the geometric identifications are accompanied by non-trivial S-dualitytransformations. They were recently used by Garcia-Etxebarria and Regalado toprovide the first construction of four dimensional N=3 superconformal theories.In this note, we classify the different variants of these N=3 preservingS-folds, distinguished by an analog of discrete torsion, using both a directanalysis of the different torsion classes and the compactification of theS-folds to three dimensional M-theory backgrounds. Upon adding D3-branes, thesevariants lead to different classes of N=3 superconformal field theories. Wealso analyze the holographic duals of these theories, and in particular clarifythe role of discrete gauge and global symmetries in holography.  In the main part of the paper, certain properties of cohomology groupsassociated to the S-folds were conjectured and used. This arXiv versionincludes an appendix written by Kiyonori Gomi in 2023 providing the proofs ofthe required properties using the technique of Borel equivariant cohomology,whose brief review is also provided.

S-folds and 4d N=3 superconformal field theories