- "You may burn us, my lady, but you will not bend us, break us or make us bow. This is Dorne, you are not wanted here. Return at your peril!"
- ―Meria Martell
With the peaceful surrender of House Stark and House Arryn, Dorne was the only part of Westeros that remained unconquered. Queen Rhaenys Targaryen flew her dragon Meraxes to Sunspear to demand Dorne's surrender. Princess Meria of House Martell refused, warning her to "return at her peril".
Confident she could subdue the Dornish with the same tactic they had used in the Field of Fire, the Targaryens returned to Dorne with their dragons. However, the Dornish refused to meet the Targaryens in the field, instead resorting to guerilla tactics. Rather than another swift conquest, it became a prolonged war with immense Targaryen casualties. Lord Joffrey Dayne invaded The Reach and led an army to Oldtown, the city where Aegon's crowning took place. Both Rhaenerys and her dragon died in the conflict. Aegon I eventually agreed to let Dorne remain independent, ending the Wars of Conquest.
More than a century later, King Daeron I Targaryen tried to finish the work of his ancestors and initiated the Conquest of Dorne. Unlike Aegon I, he actually managed to occupy Dorne, but like his ancestor, he soon found himself facing a prolonged guerilla war. The four-year war cost the lives of 60,000 Iron Throne men, and eventually also Daeron I's life. House Martell would remain the sovereign rulers of Dorne until it peacefully joined the fold through marriage.
In the books
Later historians often speak of the "War of Conquest" as beginning with Aegon's Landing at the mouth of the Blackwater in 2 BAC and neatly ending with the Field of Fire and Aegon's subsequent crowning by the High Septon in Oldtown (which marked the first day of the year 1 AC under the new Targaryen calendar system). Considerable fighting continued with Dorne and the Iron Islands, however, leaving some to refer to this entire time period as the "Wars of Conquest".
While House Hoare and most of its armies were destroyed at the Burning of Harrenhal and subsequent revolt of the Riverlands, the Targaryens did not immediately conquer the Iron Islands themselves: with so many of their forces destroyed they were no longer an immediate threat to Aegon's campaigns on the mainland, so he focused on his remaining conquests leading up to the Field of Fire. Moreover, the ironborn had been left leaderless, and their meager remaining warriors actually started fighting themselves to try to seize power. It wasn't until 2 AC that the Targaryens invaded the Iron Islands in force, promptly winning due to their dragons.
Dorne, meanwhile, was an entirely different matter altogether, with its armies still intact and defiantly relying upon defensive warfare behind its mountains and deserts. Realizing how difficult this would be, the Targaryens didn't mount a full scale invasion of Dorne until 4 AC.