Não há muitas menções sobre o que aconteceu com os dragões, mas aqui está o que eu encontrei:
Dragons were created by Morgoth. Many of them died at the War of Wrath (at the end of the First Age. That is why there aren't a lot of dragons in the Thirds Age)
(J. R. R. Tolkien, Christopher Tolkien (ed.), O Silmarillion):
During the War of Wrath, Morgoth unleashed a new terror upon Middle-earth -- the winged dragons. Chief among these was Ancalagon the Black. Eventually slain by Eärendil the Mariner, Ancalagon's fall crushed the towers of Thangorodrim. Many of the dragons were destroyed in the War of Wrath but some fled and survived into the later Ages.
O maior e mais temido (mencionado) dragão que viveu na Terceira Era foi Smaug.
Gandalf disse a Frodo:
(J. R. R. Tolkien, O Senhor dos Anéis, A Sociedade do Anel, "A Sombra do Passado")
[...] there is not now any dragon left on earth in which the old fire is hot enough [to melt the Rings of Power]
indicando a presença de outros dragões menores.
(J. R. R. Tolkien; Humphrey Carpenter, Christopher Tolkien (eds.), As Cartas de J. R. R. Tolkien, Carta 144, (datada de 25 de abril de 1954))
O que sabemos dos Apêndices do Senhor dos Anéis é que depois da Primeira Era, os Dragões apareceram para o norte para Lixo Norte .
It would appear that the dragons fled to the Northern Waste, far from the lands of Men and Elves. Over the centuries, the race of dragons continued to breed and repopulate, particularly in the Withered Heath, an area in between two spurs of the Grey Mountains.
(J. R. R. Tolkien, O Senhor dos Anéis, Apêndice B, "A Terceira Era", Apêndice A, "O Povo de Durin"):
In the late Third Age the dragons of the Withered Heath, stirred by the return of Evil, began to harass the Northmen and make war with the Dwarves around the year T.A. 2570 (Dáin I and Frór of Durin's folk were killed by a great cold-drake in 2589). It was perhaps in these wars that dragons swallowed four of the Seven Dwarf-rings.
Aqui está o que sabemos sobre Smaug (do Hobbit):
The most fearsome dragon of the Third Age was Smaug, who laid waste to the Dwarf-realm of Erebor and the nearby town of Dale. This devastated the area and sent Durin's folk into exile. Smaug remained in the abandoned halls of the Lonely Mountain for many years until the coming of Thorin and Company and their "burglar", the hobbit Bilbo Baggins. This began a chain of events that led to Smaug's death at the hands of Bard the Bowman.