Some significant nonchaotic behaviors of the lattices of coupled logistic maps are analyzed. In particular, the review concerns the organization of cycles for small coupling and the fundamental role played by heteroclinic cycles and quasiperiodic traveling waves. Moreover, we point to the existence of a few elementary cycles the stability of which determines that of almost all nonchaotic structures of any size, in particular for high nonlinearity and medium and large coupling. This allows an approximate prediction of which attractors can occur for given parameter values.