River Song, previously a doctor and eventually a professor, was a mysterious archaeologist. She was a convicted murderer who shared a close relationship with the Doctor. Significantly, she knew the Doctor's true name, which he told her at some point in his future for reasons as yet undisclosed. She would often greet him by saying "Hello, sweetie", whether it be as a message or in person.
Although the Tenth Doctor's first meeting with her ended in her physical death, they had many adventures, from the Doctor's perspective, afterwards. Like Melanie Bushand Charley Pollard before her, she was an example of the rare companion whose timeline was not synchronous with that of the Doctor.
River Song was an archaeologist already very familiar with the Doctor when they first met, from his perspective, in the 51st century. She was, in fact, a future companion of his who would (or, from her perspective, had) travelled with him on and off. According to her, he came to trust her implicitly, though at the start of their relationship, from his perspective, he was generally suspicious of her and occasionally almost hostile to her. Prior to her encounter with the Weeping Angels, she was imprisoned in the Storm Cage Holding Facility for the murder of someone she described as, "A very good man. Best man I've ever known". (DW: The Time of Angels / Flesh and Stone)
In the 52nd century, an imprisoned River Song received a phone call from Winston Churchill, who was calling from 1941, attempting to reach the Doctor. She escaped from the prison after kissing a new guard whilst wearing hallucinogenic lipstick, and stole an old Van Gogh painting, which foretold a deadly prophecy of the TARDIS exploding. Liz 10, the owner of the painting at that time, confronted her about her intentions. River then made a deal at a black market for a Time Agentvortex manipulator that she used to travel back in time to the oldest planet in the universe. There, she wrote a message on the cliff face to the Doctor, telling him to go to the Roman era of Earth. Once there, she disguised herself as Cleopatra and used her hallucinogenic lipstick to fool the Romans.
River realizes she's trapped inside the TARDIS. (DW: The Pandorica Opens)When the Doctor arrived, she explained the situation and showed him the painting of the TARDIS exploding. They then proceeded to Stonehenge where they searched for the Pandorica. They located it, but then discovered that Stonehenge was transmitting a signal, bringing in Daleks, Cybermen, Sontarans, and many other alien races. The Doctor managed to convince the assembled aliens to back away for a time, and sent River to bring the TARDIS to him. However, the TARDIS brought River to Amy's house, on the 26th of June, 2010. Looking around, River discovered burn marks scorched into the garden by alien craft and the front door broken and detached from the house. She also found a book on Romans and another on Pandora's Box in Amy's room, the book on Romans containing the exact likenesses of the Roman troops in 102 AD. Returning to the TARDIS to inform the Doctor, River then told him the date. Realising the TARDIS was about to explode, the Doctor urged River to escape. Instead, she found herself trapped as the TARDIS came under control of an unknown external force and started to explode. (DW: The Pandorica Opens)River gives the Doctor an ambiguous "yes" about her marital status. (DW: The Big Bang)
Inside the TARDIS, River was stuck in a time loop of the first moments of the TARDIS explosion. The Doctor, using the vortex manipulator River had acquired, was able to retrieve her and bring her back to 1996, where she mercilessly killed the Dalek responsible for shooting the Doctor. After the Doctor restored the universe to life, erasing himself in the process, River left her now blank TARDIS-shaped diary with Amy Pond on her wedding dayin 2010. This triggered Amy's memories of the Doctor, thus reviving him. When River and the Doctor parted ways, she apologised for something in his future and answered "Yes" to a question she interpreted both as "Will you marry me?" and "Are you married?" before teleporting off with the vortex manipulator. Before she left however, she told the Doctor he would know who she really was very soon, but apologized, warning him that that would be when everything would change. (DW: The Big Bang)
Some time after these events she was imprisoned once again, but in an earlier century. She remained so until she was put under Father Octavian's custody. If she were to carry out the mission assigned to her she would earn her pardon. (DW: The Pandorica Opens)
While working for Father Octavian, Doctor River Song was sent to be a guest on the Byzantium ship. She infiltrated the lower levels to find the Home Box, into which she carved the message "Hello Sweetie" in the Old High Gallifreyan language to attract the Doctor's attention. About 12,000 years later the Eleventh Doctorfound the Home Box in a museum.
The Doctor used the Home Box to see who had written the message. Seeing her being pursued by guards and hearing her saying the co-ordinates of her location, he went to rescue her. She then jettisoned herself from an air lock, where the waiting Doctor and Amy Pond picked her up in the TARDIS. River then showed exceptional TARDIS flying skills, landing it next to the Byzantium, which had crashed into the Maze of the Dead. Along with the Bishop and his Clerics, River accompanied the Doctor and Amy into the maze. She made it out of the maze alive along with Amy and the Doctor, to be imprisoned once more, but she thought she might have done enough to earn a pardon. She then mentioned to the Doctor that they would meet at the Pandorica. (DW: Flesh and Stone)
River's expedition to the Library was financed by Felman Lux and was chartered to discover what disaster had happened there, 100 years previously, when a cryptic final message was sent: ">>> 4022 saved. No survivors <<<". When her team ran into trouble during the expedition, River summoned the Doctor by sending him a message via psychic paper. The Tenth version of the Doctor who turned up had not yet met her. Though bewildered, he agreed to help River and her team.
River and the Doctor appeared to be very close, even considering the strong bonds the Doctor forms with his companions. She still had her TARDIS-shaped diarydescribing her travels with the Doctor. She carried an upgraded version of the tenth incarnation'ssonic screwdriver which a later Doctor had given her.
As the situation in the Library deteriorated, River realised she must prove to the Doctor that she was, or would become in his case, someone he trusted completely and with profuse apologies, she whispered his true name into his ear. The Doctor was stunned by this information, and River succeeded in winning his trust.River, moments before her death. (DW: Forest of the Dead)
River physically died when she chose to sacrifice herself in the place of the Doctor in order to save Donna and the other 4022 people trapped in the Library's computer. She pleaded with him to not try and change history, not to change one moment of what is to come for them. After saving all the trapped people, the Doctor realized his future self must have given her his sonic screwdriver for a reason. Inside he found a data chip called a "Neural Relay", which contained River's Data Ghost. The Doctor saved her by uploading her into the virtual world contained in The Library's data core. CAL, the library's data core, controlled by a human girl (Charlotte Abigail Lux, or CAL) wired into its mainframe, also managed to save the data ghosts of all of River's dead archaeological team, in the process giving her company in the virtual world. River would go on to take care of the two virtual children and Charlotte in the virtual world, reading out her diary of her life with the Doctor and how "everybody lives" when the Doctor comes to call. Her story ended with her telling the sleeping children sweet dreams. (DW: Silence in the Library/Forest of the Dead)
The Doctor would later remember her death when Davros reminded him of the people who had died in his name. (DW: Journey's End)
River at times enjoyed showing off her knowledge in front of the Doctor, particularly knowledge of him and his personality. She would often use the word "spoilers" in reference to her knowledge of the Doctor's future, or his knowledge of her future. River was also persistent at keeping secrets from the Doctor, especially anything which could change the future. This included keeping her personal diary from him.
River showed that she cared about other people, such as the Doctor and her team in The Library, but could also demonstrate signs of mercilessness, such as killing the Stone Dalek after it had shot the Doctor and begged for mercy showing a fear of her when it learned who she was. River Song could be cheeky, but was serious when the situation called for it.
River Song's relationship with the Doctor is currently a subject of mass speculation, but with each consecutive meeting so far he appears to become more trusting of her.
When the Tenth Doctor first met her she appeared to be someone he knew intimately (something she demonstrated when she stroked his hair and knew, just by his eyes, that he was much younger than "her Doctor" - hinting to some form of loving relationship) and seemed hurt when he confessed to not knowing her. Throughout the stories she made continual references to how well she knew him, even going so far as to tell the Doctor his real name to make sure she had his full trust - she described herself as someone who one day he would "trust completely" and that she herself would "trust that man to the end of the universe" (adding that they'd actually been). When she sacrificed herself to save the people in the Library (and spare the Doctor of doing it himself) the Doctor said there was only one time he could ever tell her his name (indicating that the Doctor himself, though unsure of details, does have an idea of who she is to him) to which she simply answered "shhh... spoilers" before commencing the uplink that would end her life. (DW: Silence in the Library/Forest of the Dead)
Upon the Eleventh Doctor's first meeting with her, he at first expressed resent at the idea of his own future being set in stone but later warmed to River after her part in saving Amy (when she fixed the teleport) and after musing on the idea that "time can be rewritten." During the same episode, when Amy asked if she was the Doctor's wife and when asked by River if "it could by anything that simple with the Doctor" answered "yes" River refused to confirm it, simply saying Amy was "good." Later aboard the primary flight deck of the Byzantium, when the Doctor said he "could bloody kiss [her]" River responded with "maybe when you're older" - again hinting to a romantic relationship in their future once more (and possibly ruling out the idea that she is in any way related to the Doctor as some have speculated). (DW: The Time of Angels, Flesh and Stone) By the end of the story he appeared to have at least some anticipation for their next meeting.
Upon their second meeting the Eleventh Doctor appeared accustomed to River's style of calling him and even smiled when he saw her "graffiti" on the oldest cliff face in the universe on Planet 1. He trusted River enough to send her back to the TARDIS to bring it to the Pandorica and by the end of the two parter they parted on good terms with the Doctor considering the question of her identity more of a game to him now and something he could wait to find out (it should also be noted that this was the second time the Doctor had the chance to read River's diary but chose not to). In the same story, when the TARDIS was seen to be exploding, River Song was heard repeated saying the phrase "I'm sorry my love" - presumably to the Doctor - again hinting a romantic future relationship with the Doctor. (DW: The Pandorica Opens, The Big Bang)
River blows a kiss. (DW:The Time of Angels)*River Song seemed to take shorter trips with the Doctor, living a more or less normal life between trips, rather than having one extended trip with the Doctor and never seeing him again after parting ways.
River Song was a skilled TARDIS pilot (with very different methods from the Doctor). She mentioned that she "learned from the best," but implied that her teacher was not the Doctor. (DW: The Time of Angels) Later, however, she revealed that the Doctor had indeed taught her. (DW: The Pandorica Opens) The inconsistencies of these statements may refer to different incarnations of the Doctor. It is also possible that she was just teasing the Doctor. Alternatively, since the statement that her teacher wasn't the Doctor was actually later from her point of view than the other statement, it's possible another teacher had taught her in more detail after the events of The Pandorica Opens.
Song appeared to have knowledge of Time Lord anatomy, including knowledge of regeneration. She indicated that destruction of both of the Doctor's hearts could be a circumstance after which regeneration was not possible. (DW: Forest of the Dead)
River possessed pictures of all the Doctor's incarnations (DW: The Time of Angels), though she did not know the order of the Doctor's incarnations.
River was sentenced to prison for murdering a man described as "a hero to many" and whom she called "A very good man. Best man I've ever known". Octavian felt that the murder victim's identity was something that the Doctor "really [didn't] want to know." (DW: Flesh and Stone)
Before his death, Octavian told the Doctor that "You think you know [River], but you don't. You don't understand who or what she is." (DW: Flesh and Stone)
It is unknown precisely where in time and space River Song originally came from. As the Doctor has left previous companions in different parts of the universe from where he originally found them, it is possible that River was not originally from the 51st century at all.
In a recent interview by Digital Spy, Steven Moffat made a passing comment to the effect that River Song might turn out to be very different from people's presumptions as to who she really is.