The showdown between The Hound and Brienne of Tarth, along with Tyrion Lannister's escape from King's Landing, were major events that occurred in the fourth season finale of "Game of Thrones." Another significant event in the finale was Bran Stark being rescued by the Children and then being taken to an old seer, who turned out to be the Three-Eyed Raven.
In the Riverlands, Arya Stark and The Hound stumbled upon Brienne and her squire Podrick Payne. A brutal fight ensued between The Hound and Brienne, ending with Brienne throwing The Hound down a side of a cliff. Brienne looked for Arya, but could not find her. Arya appeared and checked on The Hound, who appeared to be fatally wounded. The Hound begged Arya to put him out of his misery, but she refused, took his money and left him to die.
Tyrion, who had been sentenced to death for allegedly killing Joffrey Baratheon, escaped captivity with the help of his brother Jaime Lannister. As Tyrion escaped through a passageway, he saw his ex-lover Shae in his father Tywin's chambers. Tyrion strangled Shae to death and soon afterwards used a crossbow to kill Tywin. Varys smuggled Tyrion onto a boat and they left King's Landing together.