Since the very beginning of Game Of Thrones, there have been whispers of Aerys II Targaryen, the "Mad King" of Westeros who died a few years before the events of the show. While the show is far past the events that led to the Mad King's demise, his legacy still lives on in the show in various ways, and being familiar with what happened to the Mad King on Game of Thrones could help viewers predict how the storylines of two of the show's central character's come to a conclusion.
The only glimpse that Game of Thrones fans have gotten of the Mad King was during the Season 6 episode "Blood of My Blood" where he's seen yelling "Burn them all!" from the Iron Throne. But the best example of just who the Mad King was comes from the Season 3 episode "Kissed By Fire," in which Jaime Lannister explains to Brienne why he chose to kill Aerys and become the "Kingslayer" people know his as today.
According to Jaime, King Aerys had grown a habit of burning his enemies alive, which led to him becoming an extremely unpopular king. The Lannister family sought to end his reign of terror by sacking King's Landing, and Jaime, who had been serving as a guard of the King, suggested that Aerys surrender peacefully to his father's army.
Instead of surrendering peacefully, Aerys had a plan to combat the Lannister forces. Aerys sought to invite the Lannisters into the city and then blow the entire city up. As Jaime tells Brienne:
"He had his pyromancer place caches of wildfire all over the city: beneath the Sept of Baelor, the slums of flea bottom, under houses, stables, taverns, even beneath the Red Keep itself ... he turned to his pyromancer, 'Burn them all,' he said. 'Burn them in their homes, burn them in their beds.'"
Jaime killed Aerys' pyromancer and then stabbed the Mad King in the back to save King's Landing from meeting a fiery fate. If this description of someone who burns their enemies rings a bell, it certainly isn't ringing one that's in the Sept of Baelor because that building ceased to exist when Cersei took the wildfire that Aerys had placed beneath the Sept of Baelor and blew it up.
There are two women vying for control of Westeros who seem to love burning things just as much as The Mad King — if not more so. One of them is his daughter, Daenerys Targaryen, whose cries of "Dracarys" have started to earn her a reputation as a brutal, unforgiving leader. It seems the apple doesn't fall far from the tree, but it's hard to imagine that people would be willing to tolerate a ruler whose blood-and-fire approach to diplomacy could end in the ashes of many Westerosi citizens.
The other woman who seems to be following in the Mad King's footsteps is Cersei Lannister, who in the Season 6 finale "Winds Of Winter" follows through on the Mad King's plan, partially, and commits a terrorist act on her own city to kill most of the Tyrell house and almost all of the Sparrows of the Faith of the Seven. As if that wasn't enough, Cersei seems to be willing to risk letting the White Walkers and the army of the dead walk south without interruption so that neither Daenerys or Jon Snow can challenge her seat on the Iron Throne. It seems in her mind that losing the Iron Throne and staying alive is a worse fate than dying atop it.
If these two are willing to rule the way the Mad King ruled, they may also die the way the Mad King died, on the wrong end of a sword from someone who would prefer to not see Westeros burned to an ash. There's already suspicions that Jaime the Kingslayer will become Jaime the Queenslayer and kill his own sister to stop her from letting all of Westeros fall into the icy palms of the Night King.
If he's not interested in killing his sister and the mother of his many deceased offspring, he already has experience killing a Targaryen — who is to say he won't kill another? He's seen the damage that a dragon can do, and once the White Walker threat has been eliminated he may take it on himself to kill the woman whose dragons could burn King's Landing to the ground if she wanted. If you want to know what happens in the future in Game of Thrones, the best place to start looking is the past.