The spatial conformation of the chromatin in the nucleus of animal cells is highly organized at several levels. Little is known, however, about its evolution, mainly because of the limited number of species in which chromosome conformation experiments have been performed.In the context of the FAANG pilot project FR-AgENCODE, we realized Hi-C experiments on tissue samples from three livestock species (pig, goat and chicken livers) and integrated the resulting data with those from other assays on the same samples (RNA-seq and ATAC-seq). Within each species, general features of the genome organization -like Topologycally Associating Domains (TADs) and genome A/B compartments- could be identified. They display functional properties in line with previous reports on other model organisms like human and mouse. By comparing these structural features across species we further investigated the connections between genome conformation and function. In particular, we found a significant conservation of the gene distribution among active and inactive compartments of the chromatin throughout evolution. In addition, comparing 3D interaction profiles of orthologous regions showed that conserved TAD boundaries have stronger insulation properties than recent ones, supporting a specific selective pressure on structural components of the genome. Accordingly, various blocks of synteny contain conserved elements from several levels of organization that contribute to the gene regulatory network: sites of chromatin accessibility, orthologous genes, TADs and A/B compartments. Altogether, these results shed new light on the conservation of genome conformation from birds to mammals and illustrate the various degrees of connection between genome structure and function.