The trial by combat is a staple of A Song of Ice & Fire book series, used to determine the guilt or innocence of a subject by means of a violent duel, often to the death. In the main series, these trials mainly consist of participants electing champions, allowing great warriors a chance to do battle on behalf of their party.
Surprisingly, there are only three trials determined by combat in the main Game of Thrones series, but there are several noted in the history of House Targaryen in the centuries prior. These are documented in books like Fire & Blood, A World of Ice & Fire, the Dunk & Egg novellas, and in ing conversation and lore mentioned in the main novels. In some cases, people demanded a "trial by seven," which meant seven combatants fighting on each side in honor of the Faith of the Seven. This was an Andal tradition brought to Westeros.
13 King Maegor I Targaryen vs. Damon Morrigen - Trial of the Seven
Winner: Maegor I Targaryen
King Maegor I, also known as Maegor the Cruel, was the third king of Westeros, known for his aptitude for violence and ruthlessness against his enemies. Maegor claimed the Iron Throne after his brother Aenys died, bringing his mother, Dowager Queen Visenya, to King's Landing to help him take control of the capital. In the capital, she welcomed anyone who would oppose his rule to face him in combat. What resulted was a trial by seven, with Maegor and Damon Morrigen leading their respective sides. Maegor was the sole survivor of fourteen participants.
Both Maegor and Aenys were sons of Aegon the Conqueror, but Maegor was Queen Visenya's son, while Aenys was born to Queen Rhaenys.
12 Harrold Langward vs. Gyles Morrigen
Winner: Gyles Morrigen
King Maegor I died a mysterious death in Targaryen history, and his heir, Jaehaerys I Targaryen, gave all of Maegor's Kingsguard a choice between ing the Night's Watch or death. While the rest ed the Watch, Ser Harrold Langward demanded a trial by combat. He was opposed by young Ser Gyles Morrigen, who defeated Langward and ed Jaehaerys' Kingsguard as its Lord Commander.
11 Braxton Beesbury vs. King Jaehaerys I Targaryen
Winner: Jaehaerys Targaryen
King Jaehaerys I was known for his long-lasting and relatively peaceful reign as king, but there was an instance where he chose to take part in a trial by combat in his daughter's honor. Ser Braxton Beesbury was accused of seducing Princess Saera Targaryen, and demanded a trial by combat, electing to defend himself. In their duel, Jaehaerys killed Ser Braxton with the ancestral Targaryen Valyrian steel sword, Blackfyre.

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10 Victor Risley vs. Gareth Long
Winner: Gareth Long
During the reign of King Aegon III, following the end of the Dance of the Dragons, a brief but treasonous uprising occurred in King's Landing. Victor Risley and Gareth Long were both part of this and when Gareth Long was captured, he accused Risley of being an accomplice. Risley demanded trial by combat instead of ing the Night's Watch, so Gareth Long faced him, killed him, and went to take the black at the Wall.
9 Prince Aemon Targaryen vs. Morgil Hastwyck
Winner: Aemon Targaryen
Prince Aemon the Dragonknight is one of the first Targaryens mentioned in George R.R. Martin's books, as this Kingsguard Lord Commander is renowned as the fabled ideal of a Westerosi knight, serving five kings with honor and nobility. King Aegon IV, also known as "the Unworthy," despised Aemon for his glory, especially after this trial by combat. Aemon defended the honor of Queen Naerys Targaryen, his sister and Aegon IV's wife, after she was accused of treason by Ser Morgil Hastwyck. Aemon won, relieving his sister of her crimes. This is a famous knightly event in Westeros' history.
8 Duncan The Tall's Trial of Seven
Winner: Duncan the Tall
Ser Duncan the Tall is kind of the expert on trials by combat, as he partook in three throughout his life. The first occurred early in his days, and will likely happen in the first season of the A Knight of the Seven Kingdoms TV series. Ser Duncan defended a woman who was being assaulted by Prince Aerion Targaryen, who, along with his brother Daeron, demanded a trial by seven over the matter. Ser Duncan earned the help of Crown Prince Baelor Targaryen (Baelor Breakspear), who tragically died in the fight, helping Duncan live.
7 Duncan The Tall vs. Lucas Inchfield
Winner: Duncan the Tall
Ser Duncan's next trial by combat happens in the novella The Sworn Sword, one of the Dunk & Egg novellas. Duncan the Tall is serving as the sworn sword to Ser Eustace Osgray, who gets into a conflict with Lady Rohanne Webber. Rohanne is represented by Ser Lucas Inchfield, who actually manages to drown Ser Duncan during the fight, still dying in the process. Duncan is later resuscitated, making him the winner.
6 Glendon Flowers vs. Daemon II Blackfyre
Winner: Glendon Flowers
The next trial by combat occurs in The Mystery Knight, another Dunk & Egg novella, though Ser Duncan doesn't participate in this fight. The conflict occurs during a tourney, where Ser Glendon Flowers is accused of stealing a dragon egg. Ser Duncan claims that Glendon has been framed to eliminate him from the tournament, allowing Daemon II Blackfyre an easier path to victory. Their trial was held in a tournament joust, where Glendon knocked Daemon from his horse.
5 Lyonel Baratheon vs. Duncan The Tall
Winner: Duncan the Tall
Duncan the Tall's final trial by combat was far later in his life, after the events of the Dunk & Egg novellas. During the reign of Aegon V Targaryen (Egg), Lord Lyonel Baratheon attempted a rebellion after Aegon V's son broke his betrothal with Lyonel's daughter. This duel didn't actually end in death, though Ser Duncan defeated Lyonel in combat, forcing him to yield.
4 Rickard Stark vs. The Mad King
Winner: The Mad King
Rickard Stark's trial by combat is one of the most disturbing in Game of Thrones history, and it was among the catalysts for Robert's Rebellion. Rickard demanded trial by combat when his eldest son Brandon Stark was accused of crimes by the Mad King, Aerys II Targaryen. King Aerys declared fire as his champion, claiming that Rickard would be proclaimed innocent if he didn't burn before tying him to the rafters and lighting fire beneath him. Needless to say, fire won.