  We classify supersymmetric warped AdS$_3 \times_w M_7$ backgrounds of Type
IIA and Type IIB supergravity with non-constant dilaton, generic RR fluxes and
magnetic NSNS flux, in terms of a dynamic $SU(3)$-structure on $M_7$. We
illustrate our results by recovering several solutions with various amounts of
supersymmetry. The dynamic $SU(3)$-structure includes a $G_2$-structure as a
limiting case, and we show that in Type IIB this is integrable.
