She then approached Minokichi to breathe on him, but stared at him for a while and said, 

"I thought I was going to kill you, the same as that old man, but I will not because you are young and beautiful. You must not tell anyone about this incident. If you tell anyone about me, I will kill you." 

Several years later, Minokichi met a beautiful young lady, named Oyuki (Yuki = "snow") and married her. She was a good wife. Minokichi and Oyuki had several children and lived happily for many years. Mysteriously, she did not age.

One night, after the children were asleep, Minokichi said to Oyuki: 

"Whenever I see you, I am reminded of a mysterious incident that happened to me. When I was young, I met a beautiful young lady like you. I do not know if it was a dream or if she was a Yuki-onna..."

After finishing his story, Oyuki suddenly stood up and said 

"That woman you met was me! I told you that I would kill you if you ever told anyone about that incident. However, I can't kill you because of our children. Take care of our children..." 

Then she melted and disappeared. No one saw her again.

>> Lafcadio Hearn's Kwaidan: Stories and Studies of Strange Things

The Yuki Onna is one of Japan's most well-known and yet unknown yokai. There is no single story of the Yuki Onna. From dread snow vampire of the mountains to a loving bride and mother, she has played many roles over the centuries; worn many costumes. She is ephemeral as a windblown mist of snow, and as impossible to hold

Often described as a tall beautiful woman with long black hair and blue lips. She is inhumanely pale or even has transparent skin that makes her blend into the snowy landscape. She often wears a white kimono, but in other legends appears naked, with only her face and hair standing out against the snow. Despite her inhuman beauty, her eyes can strike terror into mortals. She floats across the snow leaving no footprints, in some stories she doesn't even have feet and she can transform into a cloud of mist or snow if threatened

Yuki Onna is only one of her names. Anywhere there is ice and snow in Japan you will find legends of the Yuki Onna, often called by some regional variation:

• 雪乳母 - Yuki Onba - Snow Nursing Mother
• 雪娘 - Yuki Musume - Snow Daughter
• 雪女子 - Yuki Onago - Snow Young Woman
• 雪女郎 - Yuki Joro - Snow Hooker
• 雪姉 - Yuki Anesa - Snow Older Sister
• 雪女房 - Yuki Nyobo - Snow Wife
  

Almost all tales of female snow monsters are also told as Yuki Onna stories. 

Although she is thought to be an ancient legend, the first known written account of the Yuki Onna comes from the Muromachi period (1333 - 1573). The monk Sogi wrote of his travels in Echigo province (modern day Nigata prefecture) and his encounter with a Yuki Onna.

Sogi writes that he went out of his house one snowy morning and saw a beautiful and unusual woman standing in his frozen garden. She was huge; almost 10 feet tall, with skin whiter than any human being. Although her face was young and beautiful, her hair was stark white and hung loosely about her shoulders. Her kimono was white to the point of being translucent, and was made of some magical gossamer fabric that clung to the woman's body.

Sogi attempted to speak to her, but she vanished into the snow. Discussing the vision later with a friend native to the region, Sogi was told that she was the Spirit of Snow (雪の精霊; yuki no seirei) who normally appeared during heavy snowfall. It was rare for her to appear at the cusp of spring.

Below are some more examples

From Tottori prefecture:

The Yuki Onna travels on the wind, and appears on days with a light snowfall. She walks through the town swinging a white Gohei wand, and shouting "Please give me water-hot or cold!" to anyone she meets. If you give the Yuki Onna cold water, she swells in size. If you give her hot water, she melts and disappears.

From Yamagata prefecture:

The Yuki Onna was once a princess of the lunar world, and lived on the Moon. Her life was full of luxury and indulgence-and extremely boring. She became fascinated looking at the world below, and so she snuck out one night and fell down to Earth, traveling on the snow. However, coming down was easier than going back up, and she became stuck on the Earth. On snowy nights of the full moon, she appears, pining for her old home 

From Aomori, Nigata, and Miyagi prefectures:

The Yuki Onna is a terrible yokai, that haunts the snowy forests looking to feed. She lives by sucking seiki (精気), the vital energy of the human body. She extracts the seiki by first freezing her victims to death, then sucking their souls out through their mouths. This type of Yuki Onna is particularly nasty in Nigata, where it is said she particularly prefers the seiki of children. 

Yuki Onna - The Talking Snow Woman

From Ibaraki, Fukushima, Akita, and Fukui prefectures:

This Yuki Onna has a peculiar trait-she needs to engage her victims in conversation in order to attack. When she meets someone on a dark and snowy night, she will call out to them. If the person answers her greeting, she attacks.

Unless you are from Fukushima or Ibaraki prefecture; in those case the Yuki Onna only attacks those who ignore her. And her method of killing is specific. Anyone who ignores her she grabs and throws into a nearby ravine. (A similar yokai in Fukui is called the Koshimusume (越娘), the Cross-Over Daughter.)

The Yuki Onba and Yukinko

Walking along a forest path at night in the dead of winter, you come upon a poor young mother. She is dressed in only a thin, white kimono, and desperately clutches her newborn baby to her chest. Seeing you, her face lights up with hope and she holds out her baby to you, begging for help. But you must harden your heart and walk away from this tragic scene-for if you take the offered baby, you will be frozen to the spot, trapped; a fresh meal for the Yuki Onba and her terrible little offspring, the Yukinko.

If the man refuses to hold the child, the consequences are equally deadly. The Yuki Onba pounds the man on the head in rage, and drives him into the snow like a hammer hitting a nail. Either way, the man becomes a feast for the Yuki Onba and her demon child.

One clever warrior got the better of the Yuki Onba though. Accepting the offered child, he took his short sword and held it between his teeth. As he held the child, he pulled its head closer to his dagger. Finally, when the sword was a hair's breadth from slicing the child, the trembling mother asked the man to return her child. The warrior returned the child to its frozen mother, who wept with joy. The Yuki Onba was so grateful she showered the warrior with gold and gifted him supernatural strength.

This story comes from Miyagi prefecture.

A group of samurai were on duty, guarding the borders of their lord's town. They camped in the forest, and huddled around a fire at night to keep warm. As they sat around the fire they swapped tales. One samurai said these woods were terrorized by a Yuki Onba, and that she had been seen recently. His companions laughed and chided him for believing in children's stories.

Eventually, one of the warriors excused himself and headed into the dark forest to relieve himself. As he went further into the forest, he saw the dim outline of a beautiful woman clutching a tiny baby. He approached cautiously, and saw that she was crying. The woman asked the samurai to please hold her small child and protect him from the cold. The samurai was moved to sympathy by the scene, and took the baby in his arms. To his surprise, it was colder than the snow around him, and stuck fast to his arms. He could not put it down. The child also grew immensely heavy, and the warrior fell to his knees under its weight. The last thing he saw on Earth was the woman's tears fading and a broad smile growing across her face.

The next morning, his companions found him frozen solid, clutching a giant icicle.

After other encounters with the deadly Yuki Onba, the samurai were determined to rid the town of her and set off hunting her in the forest. One warrior came across a tiny child running freely in the snow. The man was shocked, as the child was so small he came up only half-way to his knees. As he chased after the child, something incredibly happened. With each step, the child seemed to grow larger and larger. Soon it was past the warrior's waist, and then his shoulders, and then as tall as the warrior himself. Suddenly, the boy turned and grinned at his pursuer. And he kept growing. Right before the warrior's eyes, he grew to the size of a house.

Steeling his nerves, the warrior drew his sword and charged at the gigantic baby, slashing with all his might. Much to his surprise, the baby shattered into a million shards with a single blow, like a hollow ice sculpture. There was nothing left.

And, for whatever reason, the Yuki Onba and her child were never seen again in that forest.

Full Source List

• Tales of Japan: Traditional Stories of Monsters and Magic >> Illustrated by Kotaro Chiba
  
• https://hyakumonogatari.com/what-are-kaidan/
• https://en.wikipedia.org/wiki/Kwaidan:_Stories_and_Studies_of_Strange_Things
  

• https://hyakumonogatari.com/2013/12/18/yuki-onna-the-snow-woman/ 

• https://en.wikipedia.org/wiki/Yuki-onna

• https://hyakumonogatari.com/2013/12/17/yuki-onba-and-yukinko-the-snow-mother-and-the-snow-child/