Aegon the Dragon and his sisters Visenya and Rhaenys conquered the Seven Kingdoms and the rule of House Targaryen continued uninterrupted for nearly three hundred years. To keep the royal blood pure, the kings often followed the Valyrian custom of wedding brother to sister.

Below is the Targaryen Succession beginning with Aegon I, who ascended the Iron Throne after Aegon's Landing.

1
Aegon I
Aegon I, Aegon the Conquerer, Aegon the Dragon.

37
Aenys I
Son of Aegon and Rhaenys.

42
Maegor I
Maegor the Cruel, son of Aegon and Visenya.

48
Jaehaerys I
The Old King, the Conciliator, Aenys' son.

103
Viserys I
Grandson to Jaehaerys.

129
Aegon II
Eldest son of Viserys.

131
Aegon III
The Dragonbane, Rhaenyra's son.

157
Daeron I
The Young Dragon, the Boy King, eldest son of Aegon III.

161
Baelor I
The Beloved, the Blessed, septon and king, second son of Aegon III.

171
Viserys II
Younger brother of Aegon III.

172
Aegon IV
The Unworthy, eldest son of Viserys.

184
Daeron II
Queen Naerys' son, by Aegon or Aemon the Dragonknight.

209
Aerys I
Second son to Daeron II.

221
Maekar I
Fourth son of Daeron II.

233
Aegon V
The Unlikely, Maekar's fourth son.

259
Jaehaerys II
Second son of Aegon the Unlikely.

262
Aerys II
The Mad King, only son to Jaehaerys.

283
War of the Usurper
Aerys II is dethroned and killed, ending the rule of House Targaryen.
Details on Robert's Rebellion...

Unless stated otherwise, years are taken directly from the A Game of Thrones appendix.