  This article reviews some results of the SAGEX programme that have developed
in the understanding of the interplay of supersymmetry and modular covariance
of scattering amplitudes in type IIB superstring theory and its holographic
image in $\mathcal{N}=4$ supersymmetric Yang-Mills theory (SYM). The first
section includes the determination of exact expressions for BPS interactions in
the low-energy expansion of type IIB superstring amplitudes. The second section
concerns properties of a certain class of integrated correlators in
$\mathcal{N}=4$ SYM with arbitrary classical gauge group that are exactly
determined by supersymmetric localisation. Not only do these reproduce known
features of perturbative and non-perturbative $\mathcal{N}=4$ SYM for any
classical gauge group, but they have large-$N$ expansions that are in accord
with expectations based on the holographic correspondence with superstring
theory. The final section focusses on modular graph functions. These are
modular functions that are closely associated with coefficients in the
low-energy expansion of superstring perturbation theory and have recently
received quite a lot of interest in both the physics and mathematics
literature.
