By the time Game of Thrones came along, Daenerys Stormborn was all that remained of the legendary Targaryen bloodline, but House of the Dragon would depict the Targaryens at the pinnacle of their power.
The prequel series, which is based on the middle chapters of George R. R. Martin's fiction-filled history book Fire & Blood, describes how the dynasty came to civil conflict over succession issues and plants the seeds of its eventual collapse and fall in Robert's Rebellion, which is a predecessor to Thrones.
Among the dragonlords of Old Valyria, the city on the southern peninsula of Essos that served as the Valyrian Freehold's capital, was House Targaryen. The Freehold was an empire in all but name even though it had no emperor, and it was controlled by a tiny group of aristocratic families.
Additionally, the dragonlords were distinguished from ordinary Valyrians by their command of the mythical creatures described in A Clash of Kings as "fire made flesh." While the Freehold ruled Essos for ages, the dragonlords competed for dominance.
The Dragonstone rulers looked east for five generations, maintaining Valyrian traditions like sibling marriage but not considering Westeros as a need. As Aegon Targaryen and his sister-wives Visenya and Rhaenys rose to power, that situation altered.
The Seven Kingdoms of Westeros were split up and nearly always at war when these three arrived. Since he was young, Aegon had been interested in Westeros; it was at his request that the large wooden table at Dragonstone be carved in its shape without any limits being indicated.
His two boys would not have the same good fortune. Maegor, son of Visenya, was angry, unsteady, and uncompromising; Aenys, the oldest and son of Rhaenys, was a weak-willed dreamer eager to please. Their attempts to uphold the polygamy and sibling marriage practises of the Targaryens brought them into conflict with the Faith, which had grown beyond of the North and the Iron Islands of Westeros.
The Seven Kingdoms were openly in revolt because of their transgressions, and Maegor's death on the Iron Throne may have brought an end to the Targaryen line.
For more than 50 years, Jaehaerys and Alysanne reigned the Seven Kingdoms, outliving many of their offspring. Jaehaerys was left without a clear heir as a result of this, though. A Great Council of all the realm's lords was called to decide the issue and elect his successor.