River - the Strange Little Girl (played by Summer Glau)
A hyper-intelligent child of privilege, River discovered that the Academy in which she'd enrolled was more of a torture center than a school. No one knows yet what The Academy did to her before her brother Simon stole her back, because she's still too traumatized to tell them.
But she and Simon need to keep running—because the Academy wants their experiment back.
River was born with a mind of unparalelled brilliance. While Simon was smart enough to become a gifted doctor in the top 3% of his class at the finest medical institution, River was so bright she corrected her older brother's spelling by the time she was three years old, and even found mathematical errors in his textbooks. Their parents spoiled them and lavished them with luxuries, even signing River up for the most challenging of education programs: The Academy. Simon and his parents thought nothing of it; she was an exceptional child and belonged in an educational institution that could cater to her needs—but after a few months, Simon received a perplexing letter from her. There were misspellings and references to individuals who did not exist, to events which never happened. Simon theorized that the letter was written in some kind of code. His parents refused to take Simon's fears seriously, believing River was just being playful and Simon was irrationally overreacting to River's absence. They were wrong.
While her family thought she was safe, The Academy was actually a scientific installation (likely funded by the Blue Sun corporation). Simon decoded his sister's letters and set out to try to get her back, seeking someone, anyone—even those living in Blackout Zones—who could help him help his sister escape. Money changed hands, and soon thereafter Simon found himself captured by the local federal authorities—his father had to come down to bail him out. Upset that bailing his son out had tarnished his reputation, Gabriel Tam told Simon to give up his ridiculous mission, and Simon cut his ties to his parents by refusing to do so.
So.
Simon made arrangements for stealing River from The Academy, then made his way to Persephone and the Eavesdown Docks slums. There, he met with the people delivering cargo containing his sister in stasis from which she was later awakened on the Firefly Transport Serenity. Blue Sun still considers River their property, and they will stop at nothing to get River back.
Unfortunately, exactly what River is now remains uncertain. River's brain was operated on countless times while The Academy had her in its clutches, and she was treated for all intents and purposes like a laboratory animal. Simon was able to perform diagnostic tests and now knows for certain that, among other things, they cut into her brain multiple times. They even stripped her amygdala, which is a part of the brain used to control emotional responses and 'filter' incoming information. As he describes it: "She feels everything; she can't not." The R. Tam sessions, recorded at The Academy, show River's slow descent into her current state.
Precisely why they did these things to her is unknown. The most likely purpose was to turn her into a sort of assassin or spy. However, since the series was unceremoniously cut short, any answers to the uncertainties of motivation are assumptive. We do know to a degree of what she is capable. 
- River has taken a firearm and, after a brief glance to ascertain their position, closed her eyes and killed three men with three shots.
- She has snuck on board a bounty hunter's ship without the bounty hunter's knowledge and thwarted his attempts to take her back to Blue Sun.
- She has shown an aptitude for extraordinary assimilation of knowledge and data recall, empathy, and even what appears to be some form of psychic "reading" ability.
- She has been known to make statements, sometimes cryptic and sometimes direct, which sound as if she can read someone's mind or know of events in a stranger's past which she could not have known.
- She has never shown an indication of being able to predict the future, but she can seem to extrapolate probable outcomes based on past experience and knowledge.
- She can just look at an individual and glean more about their history and personality than some could learn from a person in years of friendship. Sometimes her statements are hard to decipher because she talks in a way that she understands but others have difficulty comprehending.
- And she's a hell of a dancer.
River's state of mind is inconsistent and unpredictable. Simon Tam administers semi-regular treatments of medication that seem to improve her ability to cope with her own mental damage for short periods of time, but these treatments have side-effects which leave her physically ill. When in good spirits, River likes playing with Kaylee and occasional practical jokes with the crew. She's potentially very dangerous, however. She caused damage to Jayne on one occasion, slashing him in the chest with a large knife (but... was she attacking him or the Blue Sun logo on his shirt?).
Quotes
The intent here is to capture what River said in the series, put it all together out of context, and hopefully extrapolate clues from that. 
Serenity
- Simon... Simon. They-they talk to me, they want me-they want me to talk.
- I slept for so long... I didn't think you'd come for me.
The Train Job
- It's not relevant. (When asked what she was dreaming about.)
- This isn't home.
- Mid-bulk Transport. Standard Radion Accelerator core. Class code 03-K64--Firefly.
- Mal, bad. In the Latin.
- Won't stop. Won't ever stop. They'll just keep coming until they get back what you took.
- Two by two, hands of blue. Two by two, hands of blue.
Bushwhacked
- It's a ghost.
- No, I can't sleep. There's too much screaming... There was.
- I followed the voices.
- Let's go again. [About her and Simon's space walk.]
- He's coming back. [Referring to the neophyte reaver, not the Captain, as is assumed.]
Shindig
- These are the ones that take you. They're the ones reaching and doing. They're the ones. ...everywhere, and the hands go everywhere... when you brush your teeth and, and nobody said anything... And then they come.
- There are tiny... Pull!
- (cockney accent) Sure, I got a secret. More than one. Don't seem likely I'd tell 'em to you now, do it? Anyone off Dyton Colony knows better than to talk to strangers. You're talking loud enough for the both of us though, ain'tcha? I've known a dozen like you. Skipped off home early, running graft jobs here and there. Spent some time in the lock down, but less than you claim. And you're what? Petty thief with delusions of standing? Sad, little king of a sad, little hill. ...Not really. Call me if anyone interesting shows up.
Safe
- (as a child) We're in trouble. We got cut off! ...Our platoon, Simon. We got outflanked by the independent squad, and we're never gonna make it back to our platoon. We need to resort to cannibalism. ...They got lost. We're gonna have to eat the men.
- (as a child) I learned it all.
- (as a child) That's wrong. ...No, the book is wrong. This whole conclusion is fallacious.
- (as a child) They [the independents] were using dinosaurs.
- (as a child) Simon lost his head in the heat of battle.
- No! No, I don't wanna go back to the... It's not okay! You can't just dig into me, shove twenty needles in my eyes and ask me what I see!
- No rutting tests? Stupid son of a bitch, dress me up like a gorram doll!
- (To Mal) You're not him? (chinese) Liou coe shway duh biao-tze huh hoe-tze duh bun ur-tze. (Stupid son of a drooling whore and a monkey)
- The human body can be drained of blood in 8.6 seconds given adequate vacuuming systems.
- Little soul, big world. Eat, sleep, and eat... Many souls.
- They weren't cows inside. They were waiting to be, but they forgot. Now they see the sky and they remember what they are.
- We're lost... lost in the woods.
- I took you away from there. I know I did. You don't think I do, but... I get confused. I remember everything. I remember too much, and... some of it's made up, and... some of it can't be quantified, and... there's secrets. But I understand. You gave up everything you had to find me. And you found me broken. It's hard for you. You gave up everything you had.
- You need to eat. Keep up your strength. We won't be here long. Daddy will come and take us home. And I'll get better. I'll get better.
- They are. Except they're poison...
- He believed her! Made a face.
- Her sister got killed. Mother got crazy, killed the sister. That one lived. Ruby doesn't talk. Her voice got scared away.
- I hear crickets.
- Yes. You're in charge. Ever since the old Patron died. He was sick. But he was getting better. You were alone in the room with him.
- Post-holer. Digging holes for posts.
- Time to go.
Our Mrs Reynolds
- Two by two. Everyone has a match, a mate, a doppler. I love you. (Explaining to Simon why she wants to marry him.)
- The captain took a wife...
- We'll take care of each other. I'll knit. You don't love me.
- (To Saffron:) You're a thief.
- Now we have to be married. [Pats pillow on belly.] I'm in the family way.
(These lines appear in a deleted scene from the DVD collection but not in the version broadcast on television.)
Jaynestown
- Bible's broken; contradictions, false logistics. Doesn't make sense.
- So we'll integrate non-progressional evolution theory with God's creation of Eden. 11 inherent metaphoric parallels already there.
- Eleven. Important number. Prime number. One goes into the house of eleven eleven times, but always comes out one.
- Noah's ark is a problem. We'll have to call it "early quantum state phenomenon." Only way to fit five thousand species of mammal on the same boat.
- I tore these out of your symbol and they turned into paper. But I want to put them back.
- They say the snow on the roof was too heavy. They say the ceiling will cave in. His brains are in terrible danger.
- No! I can't. Too much hair.
- It'll still be there, waiting.
- Just keep walking, preacher-man.
Out Of Gas
- "Day" is a vestigial mode of time measurement based on solar cycles. It's not applicable... I didn't get you anything.
- Fire... Fire! Fire!
- Don't be afraid. That's what it says. Don't be afraid... But you are afraid. You're afraid we're going to run out of air. That we'll die gasping. But we won't. That's not going to happen... We'll freeze to death first.
Ariel
- He looks better in red.
- You're going to suspend cerebral, cardiac, and pulmonary activity in order to induce a proto-comatose state. ...I don't wanna do it. ...I don't wanna go to that place. I don't want to die.
- Copper for a kiss. (As she sits up after her 'sleep', speaking to Jayne, who's sold them out.)
- They're doing it backwards; walking up the downslide.
- They're going to die. ...He is. ...He's not going to help him right.
- (to Simon re: hospital) Where you should be.
- You have to help him. ...He's killing him!
- No, no, no, they come out of the black. They come when you call.
- (to Jayne) Your treasure's in the sand.
- No, no, I can't go back. I don't wanna go back.
- (To Jayne, after they all get caught by the Feds.) They took Christmas away... Came downstairs for the shiny presents. They took the tree and the stockings. Nothing left but coal... Don't look in the closet, either. It's greedy. It's not in the spirit of the holiday.
- Doesn't matter. They're here.
- Two by two. Hands of blue. Two by two. Hands. Blue. Two.
- Almost there. Almost there. Almost there. There.
- They come out of the bike. They come when you call.
War Stories
- I threw up. ...Going. Going back like. Apple bits coming back up. Chaos.
- Played with Kaylee. Sun came out, and I walked on my feet and heard with my ears. I ate the bits, the bits stayed down, and I work. I function like I'm a girl. I hate it because I know it'll go away. The sun goes dark and chaos has come again. Bits. Fluids. What am I?!
- I threw up on your bed.
- Can't look. Can't look. (Doesn't look, just shoots the guys.)
- __No power in the 'verse can stop me.__
Trash
- She's a liar.
- She's a liar and no good will come of her.
- Jayne is a girl's name.
- (To Jayne) Also, I can kill you with my brain.
(We don't know if "the liar" refers to Saffron or to Jayne. ["Jayne is a girl's name."])
The Message
- My food is problematic.
- You are such a boob. (To Simon when he asks about the crate, after striking out with Kaylee again)
- I'm very comfortable. (Lying on Tracey's coffin.)
- 1001. 1002...
- Shh. I'm counting between the lightning and the thunder to see if the storm is coming or going. 1005...
Heart Of Gold
- It's starting! (Referring to Petaline's labor.)
- Mine too. (Her first presence at a delivery.)
- Who do you think is in there?
- A boy. Healthy.
Objects In Space
- It's just an object. It doesn't mean what you think.
- She understands. She doesn't comprehend.
- No touching.
- It's getting very, very crowded!
- You're wrong, Early. You're wrong about River. River's not on the ship. They didn't want her here, but she couldn't make herself leave. So she melted... Melted away. They didn't know she could do that, but she did.
- I'm not on the ship. I'm in the ship. I am the ship. ..River's gone. You're talking to Serenity. And Early, Serenity is very unhappy.
- Kaylee? You're afraid. ...It's okay. It's gonna be okay. Shh. I'm fine. But I need you to do something for me. Gotta be brave. ...Got tools. Something sharp. Don't be scared. I'm right here.
- Early... Mr. Jubal Early, bounty hunter. May I call you Jubal? Your mother does. I'm sorry. Did. She's gone now.
- You're a liar. I don't think your intentions are honorable. ...You hurt people.
- You're a bad liar. You crawl inside me uninvited and you hurt my crew. I see everything that passes --
- I need you to do me a favor, Captain. ...I know you have questions. But there isn't time, Captain. I need you to trust me.
- (re: dreaming) We all are.
- Don't make faces.
- He's faster than you -- all of you. And he's wearing armor.
- No touching guns!
- I'm very close to him. He doesn't even see it. Go now!
- You're not welcome here anymore, Early.
- I know it. (re: Jubal asks if she thinks he'll leave empty handed)
- Wish I didn't. (re: know Jubal very well)
- You like to hurt folk.
- It's why you took the job. Power. Control. Pain. Your mother knew. Sadness in her when she waved good-bye. She's relieved. Saw darkness in you. You're not well... Big golden retriever sitting on the lawn... Never took to you. Smelling you. The neighbors pets -- you did things to 'em. Cleaned up after... shined and polished. Everything in here gleams.
- It's very interesting... all these buttons.
- Put the gun away.
- You're not right, Early. You're not righteous. You've got issues. ...You talk too much.
- It's okay, Early. I'm going with you. ...Don't belong. Dangerous, like you. can't be controlled. Can't be trusted. Everyone could just go on without me and not have to worry. People could be what they wanted to be. Could be with the people they wanted. Live simple. No secrets. ...I'll be fine. I'll be your bounty, Jubal Early. And I'll just fade away.
- Simon!? (re: Simon attacks Jubal Early and is shot)
- Permission to come aboard?
- It's the popular theory.
- He takes so much looking after.
- I can win this.