Martha Kane Wayne was the wife of Thomas Wayne and mother of Bruce Wayne.
Martha Wayne was created by Bob Kane, Bill Finger and Gardner Fox
Golden Age / Earth-Two
Thomas and Martha Wayne were exiting a theater after seeing the Mask of Zorro. A mugger named Joe Chill stopped them and tried to steal her pearl necklace. Thomas Wayne tried to stop him but he was shot. The shock of seeing him die was too much for her and she died of a heart attack.
Bruce swore vengeance and promised to catch his parents killer someday. He would eventually find out it was Chill and went after him. It was shown that Chill has now become a racketeer 'exporting' wanted criminals along the state lines. After several failed attempts, Batman would confront Chill and reveal his real identity to him. In shock, Chill ran to his garage and asked his men to help him. He frantically tells them that he was responsible for the Batman's creation. In anger, the henchmen shot him dead.
Silver Age / Earth-One
During the Silver Age, Martha Wayne had little to do in the story besides appearing next to her husband. In this version, she gives birth to two sons, Thomas Jr and a younger Bruce Wayne. Thomas Jr. was more studious while Bruce extroverted easily displaying his emotions. One day, Thomas took his son Bruce and wife Martha to a movie.
While they were returning from a movie a mugger named Joe Chill stopped them. Chill tried to take away Martha's necklace, but Thomas tried to stop him and he was shot. Martha shouted for help but Chill shot her. Bruce shout in anger: "Y-You killed them, you killed my parents.. You!You! Something in Bruce's eye scares Chill and he runs away.
Thomas Jr. would become traumatized when he learned and he was taken to a mental hospital by his uncle Philip Wayne. Years later, Bruce find out it was Chill in a story reminiscent of the Golden Age version. Later, Bruce would learn that his parents murder was actually a hit and it was a gangster named Lew Moxon who arranged it. (a more detailed version is given in the Thomas Wayne article) Batman will go after an amnesiac Moxon in his father's bat suit and he would flee into the street where an oncoming truck will kill him.
Modern Age / New Earth
In the modern age, Martha Kane was a notorious party girl who enjoys going to country clubs, night clubs and soirees. She also comes from a wealthy family and is also a social activist. She often uses her family money to help others and aims to stop drug abuse and child trafficking. In those days, she had a friend named Celia.
One day, Martha began dating a man named Denholmn. It was revealed that Denholmn was a gangster and had past shady dealing with Celia. It revealed that Celia had been embezzling money from one of Martha Kane's charity funds and that's how she got to know Denholmn. Celia left for Greece and Martha warned her that if she returned she (Martha) would expose her crimes. Later, Martha met and fell in love with a physician Dr. Thomas Wayne. The two would later marry and give birth to their son Bruce.
The New 52 / Prime Earth
The New 52, Martha origin is very similar of the Golden Age version of Martha. Martha and her husband leave the movie theater with their young son, Bruce. As they walk down a dark alley, they are stopped by mugger named Joe Chill. Chill tired to take her pearl necklace. Thomas tired to stop Chill, But Chill shots Thomas and then shots Martha, killing the both of them.
Major Story Arcs
Modern Age / New Earth:
Death and Maidens
Batman swallows an elixir given to him by Ras Al Ghul and talks to his parents. In the dream, Bruce attempts to pick Martha's pearls up but she arrives and tells him that "it will never be better, it will never be fixed" She is shown to be somewhat like Batman in her personality. Like when she asks him to stand up for himself and keep up with her. She is shown to be generally unsupportive of Bruce's decision to become Batman. As she believes that he has forgone his chance for happiness. As she and her husband leave, they tell Bruce that even if time had lessened his grief it doesn't mean he isn't caring for them.
Martha's inspirit Bruce
As in the Death and Maiden's, Martha is shown to be caring and wants her son to be happy. She is also shown to be a strong and intelligent woman. She also created an organization with the help of Commissioner Gordon and the family butler Alfred. The goal of the organization was to stop child abuse and trafficking around the world. She believed that it would stop them from growing into criminals. Her views had also been an inspiration for Bruce as he tries to continue his mother's work. She has also taught Bruce something else in the Death and Maidens. Bruce is Batman not because he has to be but because he chooses to be.
Powers and Abilities
Martha has no powers.
Batman: Legends of the Dark Knight Annual 4: Citizen Wayne
In the story Citizen Wayne, Martha had survived the muggling and had grow feeling resentment against her dead husband for act so foolish heroic. SHe balmed him on her son Bruce for following similar heroic footsteps in an story set in the 1930s and very inspired in the Citizen Kane film.
Flashpoint: Batman Knight of Vengeance
In the Flashpoint universe, it's revealed that Martha and Thomas Wayne survived the mugging with Bruce becoming the victim. This lead to Thomas turning into Batman and Martha into the Joker. She kidnaps the son and daughter of Harvey Dent and disguises Jim Gordon's daughter like the Joker. Gordon shoots his daughter thinking it was Joker.
Eventually Batman went after the Joker and he could save in time the Harvey Dent offsprings. When Thomas chased Joker, she fell to her death.
Dark Nights Death Metal
In Dark Knights Death Metal, Batman faced a version of his mother who had become the Pearl, and fought against Batman using an armor of hulking size. Revealing her identity was an emotional impact to the main Batman.
Batman Superman: The Archive of the Worlds.
Inside the pocket dimension called Archive of worlds, Martha Wayne existed in the called World of Tomorrow. A world where Superman was the only superhero and where she and his son had survived the muggling. Decided to make the world a safer place, she make deals with the sinister Dr. Atom to introduce advanced technologies to his world, transforming it in a 1950s looking world where cellphones and personal computers already existed. However all her plans fell apart when an alternate version of her son, from the World of the Knight confronted her.