  We review our recent work on the synchronization of a network of
delay-coupled maps, focusing on the interplay of the network topology and the
delay times that take into account the finite velocity of propagation of
interactions. We assume that the elements of the network are identical ($N$
logistic maps in the regime where the individual maps, without coupling, evolve
in a chaotic orbit) and that the coupling strengths are uniform throughout the
network. We show that if the delay times are sufficiently heterogeneous, for
adequate coupling strength the network synchronizes in a spatially homogeneous
steady-state, which is unstable for the individual maps without coupling. This
synchronization behavior is referred to as ``suppression of chaos by random
delays'' and is in contrast with the synchronization when all the interaction
delay times are homogeneous, because with homogeneous delays the network
synchronizes in a state where the elements display in-phase time-periodic or
chaotic oscillations. We analyze the influence of the network topology
considering four different types of networks: two regular (a ring-type and a
ring-type with a central node) and two random (free-scale Barabasi-Albert and
small-world Newman-Watts). We find that when the delay times are sufficiently
heterogeneous the synchronization behavior is largely independent of the
network topology but depends on the networks connectivity, i.e., on the average
number of neighbors per node.
