  The discrete-time two-dimensional Toda lattice of $A_\infty$-type is studied
within the direct linearisation framework, which allows us to deal with several
nonlinear equations in this class simultaneously and to construct more general
solutions of these equations. The periodic reductions of this model are also
considered, giving rise to the discrete-time two-dimensional Toda lattices of
$A_{r-1}^{(1)}$-type for $r\geq 2$ (which amount to the negative flows of
members in the discrete Gel'fand--Dikii hierarchy) and their integrability
properties.
