Talia al Ghul was the daughter of Ra's al Ghul.
Talia is the daughter of Ra's al Ghul and one of his most loyal members of the Society of Shadows. Her mother's name, the details of her early life, training, or education have never been disclosed, not even whether or not she has siblings besides her half-brother, Arkady Duvall. She is first seen acting as her father's agent/emissary. He assigned her to hunt down a former associate of her father's named Vertigo, who went rogue. During this assignment, she encountered Batman and the two began an on again off again romance and would continued to cross paths several more times over the years. Together, they stop Vertigo's plan.
Some time later, Ra's al Ghul appears to Batman in the Batcave and informs him that his daughter was kidnapped along with Robin. This proved later to be a ruse, so that Talia could meet Batman and hopefully marry him which he refuses. After stopping Ra's' plot to "cleanse" the world, Talia prevents Batman and Robin from arresting her father and informs her beloved that they shall meet again.
Talia meets Batman yet again in order to find her father. This time however Bruce Wayne appears more infatuated with Talia even going so far to admit that he cares for her. After rescuing Ra's' from his latest plan, they part ways yet again.
Battle with Superman
Trying to save the life of her father, Talia and some henchmen rob a train holding Native American artifacts. Superman tries to stop her, but in the nick of time she finds the Shaman's Staff and uses it to subdue Superman. Combining the staff with a computer, she transfers Superman's powers to revive a dying Ra's al Ghul. Though their latest quest was successful, the two were tossed into an underground river after an ensuing scuffle with the Man of Steel and the Dark Knight.
Near-Apocalypse of '09
Although not much is known about this event, it is known that Talia betrayed her father and helped Batman stop Ra's al Ghul one last time. Around this time, Bruce and Talia dated at least once in Milan.
After the Near-Apocalypse of '09, Talia appeared to be leading a reformed Society of Shadows, having transformed it into a huge charitable corporation which sponsored numerous projects to improve the Earth's environment. She was able to keep her young, beautiful appearance by using the Lazarus Pits like the one in her residence in New Cuba. On Bruce's birthday, she reappeared to him in the Batcave offering him roasted pheasant and a chance to regain his youth.
At first, Bruce decided to undergo the Lazarus Pit treatments and as such traveled with Talia and Terry McGinnis to her estate on New Cuba. After undergoing the first treatment, Bruce regained the appearance as well as the body of a physically fit and strong man in his early fifties. Believing that the Lazarus Pit was a cheat, Bruce decided to stop the treatments, but the Society of Shadows refused to let them go and then an awful secret was revealed. After the Near-Apocalypse of '09, Talia was asked, and excepted, to be purged from her body to house the spirit of her father, effectively killing her. Talia's body was destroyed after Terry rescued Bruce and an exposed electrical cable came into contact with the Lazarus Pit, causing a massive explosion. After the explosion, Terry wanted to go back in and rescue her, but Bruce stopped him because "Whatever was in there, died years ago".
Powers and abilities
Talia is an Olympic-level athlete, having been trained since birth in many forms of martial arts. She is also quite proficient with most hand weapons. Often underestimated, but never quite forgotten, Talia is also an excellent hand-to-hand combatant.
Talia al Ghul is one of several children Ra's al Ghul has sired in his multicenturial existence, and the one he regards as the most competent. If she is like her comic counterpart, then she served as her father's second-in-command. While Ra's genuinely loves Talia, he is also an avowed sexist, and deems women inferior to men; thus her husband, not her, is to succeed Ra's. Although Talia follows her father, she is not a villain like Ra's, nor is she a hero like Batman. Her motivations are based on her love for her father and Bruce Wayne, sometimes supporting one, sometimes another.
Following the Infinite Crisis, Talia revealed to Bruce Wayne that she had borne him a son, Damian Wayne, who became the Robin to Batman's successor Dick Grayson, and later serving as sidekick to Bruce himself after he returned to the present time. Damian's decision to follow his father rather than his mother fueled Talia's enmity for the Batman family.