In fiction and in the media in general, both heroes and villains usually agree on one thing: they Wouldn't Hurt a Child.
This creed exists for various reasons: it may be because Children Are Innocent and thus should be protected and kept out of harm's way at all costs; it may be because, to Moral Guardians, seeing a child being brutalized is Harmful to Minors thus should be avoided; may also be because even writers mostly disagree with the idea of harming a child.
But sometimes, some people just don't agree with that. Or just don't care. And, they do hurt children, sometimes even kill them, in defiance of the rule of Infant Immortality. Indeed, the fact of harming a child usually comes off as the ultimate act of villainy and it takes a very specially evil character to do this.
Needless to say, this trope is almost always a Kick the Dog moment. Hence, the presence of Would Hurt A Child is often telltale of a Crapsack World or at least a Darker and Edgier series. However, it is surprisingly easy to subvert this trope simply by using Kids Are Cruel (the flip side of Children Are Innocent) and combining it with Pay Evil unto Evil. Want to go all the way? Make the children creepy or, even better, make them Enfants Terribles and, this time, hurting said children will look much less as a Moral Event Horizon crossing, even though there will always be some Moral Guardians or Papa Wolf / Mama Bear to complain. This may also be done by a Mook to establish that even the Big Badhas lines s/he won't dare cross.
In Western works, this trope is rare and almost always done off-panel when played straight (read: when the violence towards the child isn't being stopped in the nick of time and causes actual damage). In Japanese fiction, child brutality is more common.
Important Note: This trope is about adults physically hurting children, fatally or not. The following are not examples: slapping, butt-warming and general discipline from parents to their children (that's Don't Make Me Take My Belt Off), psychological/verbal abuse (very common but not that trope), abduction (only applies if the child dies or gets effectively hurt), child on child violence (that's Kids Are Cruel), exploitation for profit (that is Financial Abuse). Violence to teenagers isn't an example either since teenagers are physically stronger than children and lack the "completely defenseless" element, especially if Teens Are Monsters. However, teenagers harming children applies.
There is also an interesting Double Standard when it comes to characters applying for this trope, since they are almost Always Male. Rarely are women described as being physically violent to children. Even rarer are the instances when they actually kill them. Indeed, killing children is among the few things that permanently prevent a woman from a convenient High Heel Face Turn. Extra points if the killing is done directly and at the woman's hands (i.e. stabbing, strangling etc...). Compare Pædo Hunt, another child abuse trope where female perpetrators are similarly unheard of.
A subtrope of Kick the Dog. See also Child Hater, who hates kids but isn't necessarily physically abusive. Contrast Infant Immortality and Wouldn't Hurt a Child. Compare Kids Are Cruel, Children Are Innocent. Related to Eats Babies and Child Eater.
In one occasion, he and Catwoman took a whole rich family hostage, including a tiny Ill Girl who was in a coma. Joker was completely ready to kill the little girl in front of her father and older brother, but a VERY pissed-off Catwoman stops him and screams that this isn't what they came for.
Perhaps Joker's most notorious moment of this came during the "No Man's Land" mega-arc, where he plotted to kill all the newborns in Gotham City on New Year's Day, just so the Joker could destroy the already fragile spirit of Gotham's citizens.
So has Two-Face. Ask all the Robins.
What The Joker did in The Nail, torturing and killing Dick and Barbaralin live television. No wonder this is one of the VERY counted times Batmanbreaks his not-killing vow.
Death Of The Family: Joker has a track record for this. He certainly has no problem fighting Damian, that 10-year-old kid who beat him up with a crowbar.
Aquaman's arch-enemy Black Manta won notoriety by killing the hero's infant son.
The Scarlet Witch from The Avengers stood in awe as Pandemonium deformed and grafted her children on his own arms.
Fantastic Four villain the Wizard had absolutely no qualms about performing an experiment on Franklin Richards that had a high probabilty of causing Franklin's death. Fortunately for Franklin, the Wizard's current partner the Mad Thinker would not hear of it, dissolved their partnership and contacted the Thing, who helped him defeat the Wizard.
Winnowill in ElfQuest imprisons, humiliates and tortures the Wolfriders, but it's only when she threatens Cutter and Leetah's children that she really crosses the line. Then there's her 'relationship' with her son Two-Edge.
One of the first things the Plutonian does in Irredeemable is vaporize a child.
Carnage from Spiderman no qualms about who he kills, children included. During his rampages, child corpses are often among the dead, and in his youth he torched an orphanage.
Sabretooth from X-Men is known to attack, murder, and sometimes even eat children. In one issue, Deadpool is in a cabin with him and he opens a closet to find a little girl tied up. When he asks about this, Sabretooth replies that he's saving her for later.
Dodge from Locke And Key has no problem pushing a child under a school bus because he's figured out too much.
In Cerebus The Aardvark, this starts showing up once the Cerebus Syndrome kicks in, with the most notorious example being Pope Cerebus blessing an infant after his mother begs Cerebus to do so and then hurling it away to illustrate the point that "you can get what you want and still not be very happy."
Doctor Doom showed no hesitation in trying to murder the Hulk's young son, Skaar, and at the end of Avengers: The Children's Crusade, he kills the young Cassie Lang. On the other hand, God help anyone who tries to hurt one of Latveria's children when he's around.
In one issue of The Spirit, two bank robbers plan to kill all witnesses to their crime — including a little girl just coming around the corner. With tommy guns. note Frightened, she falls down — and survives.
In Superman, Catherine Jane "Cat" Grant's son Adam Morgan is first kidnapped by Toyman (alongside other children), then is stabbed to death when he attempted to escape from him.
My Little Pony Friendship Is Magic IDW: Queen Chrysalis has made it quiteclear that she doesn't intend on letting the Cutie Mark Crusaders leave. Not to mention a luvcat kitten was seen before Chrysalis and her soldiers took over the village.
In the Dashverse story Hot Heads, Cold Hearts and Nerves of Steel, Sombra has his minions kidnap every foal in Ponyville and several other towns so he can drain their life forces to make himself stronger.
His bushwoolie Mooks have no problem with harming foals, and the two that Pip, Alula and Dinky encounter try to eat them.
Hitman Miami: In mission (chapter) seven, 47 (the protagonist) needs to get money for his supermarket shopping. The only way is, apparently, garrotting children in the toy section and stealing their change.
Armitage III fanfic Gods And Monsters features a female protagonist who is more than willing to kill children. Even the author called this out as being hard to pull off.
Reconstructed in The Incredibles. Helen has to explain to Dash and Violet that yes, real villains would hurt a child. And they do. Then again, the kids can fight back...
The other PG Pixar Film, Up, has the villain send dogs to attack the kid there, and before he died, almost shot the kid as well. Pixar seems to tell people that villains can, and WILL, harm and kill children if they have the option.
Speaking of Pixar, a plan involving Randall and Waternoose in Monsters Inc may not have specifically aimed at killing children, but the results of the Scream Extractor seem anything but harmless.
The Great Mouse Detective: The Faux Affably Evil Ratigan is revealed - in song, no less - to have drowned widows and orphans. Near the end, during the Big Ben battle, he's shown kicking Olivia off a large gear, directly into the path of being crushed by two other gears. Luckily Basil is able to rescue her, just barely in time.
The Hunchback of Notre Dame: Judge Claude Frollo, in the first few minutes of the film, would have dropped baby Quasimodo down a well if not for the intervention of the Archdeacon, who demanded that Frollo raise the infant to atone for killing Quasimodo's mother.
Captain Hook may be Laughably Evil through most of PeterPan, but he did put a bomb in a house full of children and made a teenage girl walk the plank.. and just watch him go all Ax Crazy in the climax against Peter.
Madam Mim in The Sword in the Stone tells 12-year old Wart she's afraid she's going to have to kill him.. by turning herself into a cat while he's transformed into a little bird. It's downplayed because Merlin intervenes in time, but still.
Shere Khan in TheJungleBook is insanely devoted to slaughtering a certain human boy.
Para Norman: Agatha, a young girl who can talk to dead people like Norman can, is executed due to the town's elites coming to the conclusion that she is a witch.
And the modern day townspeople who were enthusiastically planning to BURN NORMAN TO DEATH.
Recess Schools Out shows us that the villians will attack children, though they fail for the most part.
The Grand Duke of Owls in Rock-A-Doodle, near the film's climax, asphyxiates child protagonist Edmund when the boy-turned-kitten refuses to give up, in a supremely dark example of Mood Whiplash. A combination of Disney Death and All Just a Dream makes it better, but it's still shocking.
King Candy in Wreck It Ralphtried to delete Vanellope from the game's code, and when that failed turned her into a glitch. Then when she races him, he slams into her, tries to knock her out using a dislodged gearshift and finally pushes her in front of his car intending to smash her into a stalactite. This trope may be slightly out of place due to all characters being ageless video game characters, but Vanellope was programmed as 9 years old.
In The Bible, King Herod did this in order to try to kill Jesus. Didn't work.
Earlier, Pharaoh killed Hebrew sons to keep the Hebrews from becoming strong enough that in case of a war they could join the enemy and fight against Egypt. Fortunately for the Hebrews, Moses escaped slaughter.
And God himself kills the firstborn of Egypt as one of the Ten Plagues, as well as David and Bathsheba's firstborn.
Many Greek myths punish a parent by slaughtering their children.
Gudrun from Norse Mythology kills her own children who she birthed to Attila the Hun and puts them in his food.
Stand Up Comedy
Mike MacDonald, when he and his wife were trying to conceive a child, noted in his act he was going to do his best not to hit his kids, ever—unless, you know, they were coming at him with a knife or something.
I don't wanna be one of those liberal parents who gets stabbed to death by their kid while saying, "Son, I'm sensing hostility OH GOD!"
One of Bernie Mac's punchlines in his act was threatening to beat his kids "till the white meat shows".
Denis Leary, in a larger bit about celebrity entitlement, said that he was looking forward to beating the shit out of his kids.
That's therapy for you: mowin' the lawn while cryin'.
Universal Orlando's Halloween Horror Nights implied this with their 2010 haunted maze, The Orfanage: it's heavily implied that the entire house takes place after the orphanage has burned down and you're being assaulted by the ghosts of the children, with various imagery and audio hinting at various tortures that the children were put through by their caretakers until Cindy demonstrated her abilities and burned down the orphanage with all the kids trapped inside.
Kirei Kotomine of Fate/stay night rounded up all the surviving orphans of the Fuyuki Fire and cut off their limbs before sealing them in coffins that drained their prana. They were alive and aware for the entire ten years between the two wars.
Zouken Matou has no issues with Shinji treating Sakura harshly or raping her. In fact, he has gone to significant lengths to break her psychologically.
Al Chest Breach. Played for laughs. Word to the wise: don't try talking over Al.
Ronald Dillinger, the child whom Doctor Dan (played by Al) hunted down and killed Ronald because he escaped life as a slave.
The ancient black dragon in The Order of the Stick intended to kill Varsuuvius' children, and managed to hurt them before V showed up to stop her. She notes that people don't think anything of killing dragon young, such as when V killed her (young adult) son.
Rich brings up a good point about dragon wyrmlings having stats and therefore indicting many PCs for slaughtering dragon children despite them being clearly sentient. The mother dragon's whole beef was with V killing her son.
Then again, the reason most people find hurting or killing children to be offensive (at minimum) is because children are usually helpless, defenseless, harmless innocents, four things which a Young Adult Black Dragon most emphatically is not. A young adult black dragon is a significant threat to a party of humanoid adventurers; two humanoid children are not remotely a threat to an adult Black Dragon.
Mavra Chan in Terinu was shown onscreen beating the life out of the title character (when he couldn't even fight back) and in a flashback having him set upon by bloodsucking animal as part of this Training from Hell. Then there's Admiral Blake, who sends assassins after the 15-year old boy...
In Axe Cop Babysits Uni-Baby, Dinosaur Soldier summons two cyborgs to "punch (Uni-Baby) any time she would cry or make a ruckus." This trope is subverted in that Axe Cop and his partner are not evil, and are in fact the heroes of the story. The story was originally written by a 7-year-old who makes no distinction between children and adults, and considers the act of punching someone in the face to be fairly equal-opportunity. Instead of a cyborg punching a defenseless infant, it's supposed to be a cartoon cyborg punching a cartoon baby.
Kore in Goblins murders a child in the first scene he appears in. The child in question is an orphan who's been Raised by Orcs, and to Kore's twisted perception of good and evil, anyone who potentially has sympathy for the 'monstrous races' is guilty of evil through their passiveness.
In Unsounded the Red Berry Boys, particularly their leader Starfish. They use human bodies as smuggling vessels, and are paid by the body. Starfish realizes they can stretch their product further by using smaller bodies.
In one of his Counter Monkey episodes, Spoony warns that if you ever DM a game, never put the players in a situation where they come across some orcs looking after orc babies. Because inevitably, someone won't want to kill the babies, but someone else will.
Whateley Universe: the Mafiosi in "Mimeographic" who were perfectly willing to kill a twelve-year-old boy unless the kid attempted to commit murder for them.
Also, Jared Reilley's father, who - once he decided for no rational reason that Jared might not be his son - tried to beat Jared to death with his bare hands. He only failed because Jared manifested as a mutant under the stress.