It's safe to say that some of the most of the pivotal events of Game of Thrones wouldn't have happened if not for Lord Petyr Baelish, also known as Littlefinger (Aiden Gillen). In the season 7 finale, 'The Dragon and the Wolf,' Littlefinger's years of schemes and manipulations from behind the scenes were at last laid bare and he received his just desserts from the very family he harmed the most - the Starks. The three surviving Stark children - Sansa (Sophie Turner), Arya (Maisie Williams) and Bran (Isaac Hempstead-Wright) - tried Littlefinger for treason at Winterfell in front of the bannermen of the North and the Vale and found him incredibly guilty. His demise was poetic - throat slit by Arya with the same Valyrian steel dagger he once tried to have Bran killed with.
Littlefinger had been pitting sister against sister ever since Arya returned to Winterfell. When Arya dueled Brienne of Tarth (Gwendoline Christie) and showed off her amazing skills she learned in her years studying to be a Faceless Man of Braavos, Littlefinger immediately knew she was a threat to him. Sansa seemed alarmed at how her sister had changed but from the looks exchanged between Arya and Littlefinger, he knew death had come to the doors of Winterfell, and it had come wearing Arya's face (or her wearing its). Earlier in that same episode, "The Spoils of War," Littlefinger gave Bran that very same Valyrian steel dagger as an ironic gift and Bran countered with the very words Littlefinger once said to Varys in private years ago: "Chaos is a ladder."
Bran was the key to unlocking all of Littlefinger's multitude of crimes against the Seven Kingdoms and the Stark family. As the Three Eyed Raven, Bran has been traveling through time and witnessing past events. Since returning to Winterfell, he had been privately exploring the events that led to his family's many tragedies, especially the most pivotal one that started it all: the execution of their father Ned (Sean Bean) at the end of season 1. While the audience didn't get to witness Bran's visions in this case (since we're already privy to them all as we watched the series), we know he had them and learned the truth, which he then revealed to Sansa. The proof is in the very words Sansa said that condemned Littlefinger when she accused him of all of his crimes.
LITTLEFINGER'S CRIMES AGAINST THE STARKS
When Game of Thrones began, the first major event was King Robert Baratheon (Mark Addy) and the royal family, including Queen Cersei Lannister (Lena Headey) and Jaime Lannister (Nikolaj Coster-Waldau), traveling north to Winterfell to recruit Ned Stark to be Robert's new Hand of the King. The previous hand, Lord Jon Arryn of the Vale, had died under mysterious circumstances. We now know it was Littlefinger who had Jon's wife Lysa Arryn - who'd known him since their childhood at Riverrun and was in love with Littlefinger - to poison Jon with Tears of Lys. Littlefinger then had Lysa write a letter to her sister Catelyn (Michelle Fairley), Ned's wife, accusing the Lannisters of murdering Ned.
When Bran witnessed Jaime and Cersei having sex in an abandoned tower in Winterfell, Jaime pushed Bran out of the window, putting Bran in a coma. Littlefinger then paid a cutthroat assassin and gave him his own Valyrian steel dagger to kill Bran; the goal was to make the Starks believe the Lannisters were trying to finish the job and murder their son. Ned had traveled to King's Landing so that Ned could assume the position of Hand, but after the assassination attempt failed, Catelyn herself traveled to King's Landing with the dagger to inform Ned.
In King's Landing, Littlefinger admitted the dagger belonged to him but he claimed he lost it in a bet to Tyrion Lannister, furthering his accusation that the Lannisters were acting out a plot to destroy the Starks. When Cersei arranged for Robert to be mortally wounded in a boar hunt, Ned was named by Robert as Lord Protector of the Realm, which threatened Cersei's interests in seeing her own son Joffrey (Jack Gleeson) become King as the heir to the Iron Throne. Littlefinger then conspired with the Lannisters to betray Ned; they had him arrested in a coup, imprisoned, and tried as a traitor - though the original plan was for Ned to admit treason (which he did under duress to protect his daughters), take the black and join the Night's Watch. Instead, Joffrey took it upon himself to order Ned executed, and Ned had his head cut off - which spun off Arya into her own quest to find the means to avenge her family, ultimately resulting in her becoming a Faceless Man.