  The anti-self-dual Yang-Mills equations are known to have reductions to many
integrable differential equations. A general B\"acklund transformation (BT) for
the ASDYM equations generated by a Darboux matrix with an affine dependence on
the spectral parameter is obtained, together with its Bianchi permutability
equation. We give examples in which we obtain BTs of symmetry reductions of the
ASDYM equations by reducing this ASDYM BT. Some discrete integrable systems are
obtained directly from reductions of the ASDYM Bianchi system.
