  We construct a continuous one parameter family of $AdS_4\times S^1\times S^5$
S-fold solutions of type IIB string theory which have nontrivial
$SL(2,\mathbb{Z})$ monodromy in the $S^1$ direction. The solutions span a
subset of a conformal manifold that contains the known $\mathcal{N}=4$ S-fold
SCFT in $d=3$, and generically preserve $\mathcal{N}=2$ supersymmetry. We also
construct RG flows across dimensions, from $AdS_5\times S^5$, dual to
$\mathcal{N}=4$, $d=4$ SYM compactified with a twisted spatial circle, to
various $AdS_4\times S^1\times S^5$ S-fold solutions, dual to $d=3$ SCFTs. We
construct additional flows between the $AdS_5$ dual of the Leigh-Strassler SCFT
and an $\mathcal{N}=2$ S-fold as well as RG flows between various S-folds.
