|Taken directly from Katara's relationships page.|
Katara's relationship with Zuko was complicated. They first met when Zuko invaded the Southern Water Tribe village. Their relationship was often antagonistic as Zuko continued his attempts to capture Aang. She grew to hate him when Zuko threatened to hurt her and her brother and tried bribery, using her mother's necklace. Regardless, Katara offered to help Zuko by healing Iroh only to be violently rebuffed. Later, when the two were imprisoned in the Crystal Catacombs, Katara began to trust Zuko after he revealed that he too had lost his mother to the Fire Nation, saying: "I'm sorry. That's something we have in common." She eventually trusted him to the point where she offered to heal his scar. Later, when Zuko betrayed her and the team by joining Azula, Katara was visibly hurt and her hatred for Zuko only worsened, as she took his betrayal personally.
Consequently, Katara was the most vocal of the group in not allowing Zuko to join them. When Zuko was eventually accepted, Katara confronted him, coldly voicing her distrust of Zuko and his sudden willingness to help Aang, even going so far as to threaten Zuko's life. Even long after the others in the group trusted Zuko, she still remained harsh toward him. Zuko tried to stay out of Katara's way, but eventually confronted her when Katara remained unmoved by his sacrifice even after he saved the team from Azula. This led Zuko to try and reconcile with her by offering to help Katara avenge the death of her mother. With the help of a "forgiveness speech" by Aang, their complex relationship was finally thawed out and Katara forgave Zuko.
Katara and Zuko became very good friends and grew to respect one another greatly. When Sozin's Comet arrived, Katara personally assured Zuko that Iroh would forgive him. Later, Zuko specifically asked Katara to help him overthrow Azula — an offer which she gladly accepted. When the two arrived, Azula challenged Zuko to an Agni Kai; Zuko accepted the challenge, explaining to Katara that he felt victory was feasible as Azula was noticeably "slipping". During the duel Azula fired a bolt of lightning at Katara, and Zuko rushed to her aid, becoming critically wounded after a botched attempt at lightning redirection. This forced Katara to fight Azula on her own. After disarming the princess, she immediately ran over to heal Zuko and voiced her gratefulness for saving her life. The two remained friends after the Hundred Year War.
Later on, when Katara was retaliating at Zuko's guards for attacking Aang, Zuko came along and restrained Katara for attacking his people, to the point where Katara claimed he was hurting her. After this, however, the two again seemed at ease with each other, including Katara suggesting a peaceful end to the debate between him and Aang with a meeting between them and the Earth King.