The nuclear genomes of vertebrates are mosaics of isochores, very long stretches (>>300kb) of DNA that are homogeneous in base composition and are compositionally correlated with the coding sequences that they embed. Isochores can be partitioned in a small number of families that cover a range of GC levels (GC is the molar ratio of guanine+cytosine in DNA), which is narrow in cold-blooded vertebrates, but broad in warm-blooded vertebrates. This difference is essentially due to the fact that the GC-richest 10-15% of the genomes of the ancestors of mammals and birds underwent two independent compositional transitions characterized by strong increases in GC levels. The similarity of isochore patterns across mammalian orders, on the one hand, and across avian orders, on the other, indicates that these higher GC levels were then maintained, at least since the appearance of ancestors of warm-blooded vertebrates. After a brief review of our current knowledge on the organization of the vertebrate genome, evidence will be presented here in favor of the idea that the generation and maintenance of the GC-richest isochores in the genomes of warm-blooded vertebrates were due to natural selection.