Vertebrate paleontology is a large subfield to paleontology seeking to discover the behavior, reproduction and appearance of extinct animals with vertebrae or a notochord, through the study of their fossilized remains. It also tries to connect, by using the evolutionary timeline, the animals of the past and their modern-day relatives.
The fossil record shows aspects of the meandering evolutionary path from early aquatic vertebrates to mammals, with a host of transitional fossils, though there are still large blank areas. The earliest known fossil vertebrates were heavily armored fish discovered in rocks from the Ordovician Period about 500 to 430 Ma (megaannum, million years ago). The Devonian Period (395 to 345 Ma) brought in the changes that allowed primitive air-breathing fish to remain on land as long as they wished, thus becoming the first terrestrial vertebrates, the amphibians.
Amphibians developed forms of reproduction and locomotion and a metabolism better suited for life exclusively on land, becoming more reptilian. Full fledged reptiles appeared in the Carboniferous Period (345 to 280 Ma).
The reptilian changes and adaptations to diet and geography are chronicled in the fossil record of the varying forms of therapsida. True mammals showed up in the Triassic Period (225 to 190 Ma) around the same time as the dinosaurs, which also sprouted from the reptilian line.
Birds first diverged from dinosaurs between 100 Ma and 60 Ma.
Read more about Vertebrate Paleontology: History, Paleontological Vertebrate Classification