  We study $\mathcal{N}=1$ supersymmetric AdS$_3 \times M_7$ backgrounds of
Type IIB supergravity, with non-vanishing axio-dilaton, three-form and
five-form fluxes, and a "strict" $SU(3)$-structure on $M_7$. We derive the
necessary and sufficient conditions for supersymmetry as a set of constraints
on the torsion classes of the $SU(3)$-structure. Given an Ansatz for the
three-form fluxes, the problem of also solving the equations of motion involves
a "master equation", which generalizes ones that have previously appeared in
the literature.
