  We discuss boundedness properties of certain classes of discrete bilinear
operators that are similar to those of the continuous bilinear
pseudodifferential operators with symbols in the H\"ormander classes
$BS^{\omega}_{1, 0}$. In particular, we investigate their relation to discrete
analogues of the bilinear Calder\'on-Zygmund singular integral operators and
show compactness of their commutators.
