Years before the events of Arkham Asylum, Batman and Talia would encounter each other in Metropolis which ended with them spending the night together. Talia herself appears to remember the night fondly and it is implied that this was the beginning of their love-hate relationship.
Batman: Arkham City
Talia appears in Wonder City to save Batman from her elite guard. Talia tries to convince Bruce to join her cause and begins to kiss him until she sees the damage the Titan disease has done to his face.
Batman tells Talia he wishes to become an assassin, waving off her suspicion by using his illness as his reason for this sudden and desperate change of heart. Talia leads him to the first "Trial of the Demon", and after Batman succeeds, she takes Batman to Ra's al Ghul, When Ra's tells Batman to take his rightful place as the Demon by killing him, Batman reveals he only needed Ra's al Ghul's blood, after which Talia becomes angry at Bruce's lie, though she still urges him to take his place as the Head Of The Demon.
After Ra's al Ghul is defeated, he holds Talia hostage to make the Dark Knight kill him, but this backfires when Batman manages to disarm him and Talia flees the room, angered that she had been betrayed by both Batman and her own father. Despite Bruce's betrayal, she keeps an eye on her beloved. Talia later steals the Titan antidote back from Harley Quinn for Batman and ties her to a pole in the Steel Mill.
She later appears in the Steel Mill saving Batman from Joker and offering the clown "the gift of immortality" giving him the position she had previously offered Batman. As she leaves, she shows Batman she has activated a tracker that he used to find her in Wonder City.
When Batman escapes the steel mill, his immediate goal is to save Talia from the Joker rather than stop Protocal 10 and Hugo Strange though both Oracle and Alfred Pennyworth defy his wishes, stating Batman cannot let one thousand lives die for the sake of one.
Though Batman agrees to stop Strange, he has Barbara keep a lock on Talia's signal. After Protocol 10 is stopped, the Joker broadcasts that he is holding Talia hostage in the Monarch Theatre When Batman arrives, he tries to convince the Joker to let Talia go free which distracts Joker long enough for Talia to kill him with her sword, explaining the only way to end his reign of terror is to kill him.
When Talia presents the antidote to Batman, he realizes that there were two Jokers the whole time, and Talia is shot and killed by the real Clown Prince of Crime. Talia apologizes to Batman for her mistake before dying in his arms, dropping the cure for Clayface (who had been masquerading as the Joker) to take. Batman uses Talia's sword to defeat Clayface in the final battle as well as destroy the machinery around the Lazuras Pit preventing Joker from reaching immortality.
Batman Arkham Knight
Talia's corpse was never been found. In the Evidence Room, Batman put his arm on Talia's sword as he apologize to her for unable to save her.