Miyu is the complete Holy Grail from the world in which the Class Cards originated. Her real name is Miyu Sakatsuki (朔月 美遊, Sakatsuki Miyu?) of the Sakatsuki family, the hidden descendant of the Child of God (神稚児, Kami Chigo?). She was raised carefully within the Sakatsuki residence and was given limited education, isolated from other children and relied only to her mother.
When Julian Ainsworth unleashed the black shadow in Fuyuki in the Fourth Holy Grail War, Miyu responded to the cries of the people and nullified the shadows. Shirou Emiya found her location when she activated the wish and saved her from a collapsing roof. Shirou brought Miyu back to Kiritsugu Emiya, who was thrilled to learn that she is the tool that he was looking for. Kiritsugu, Shirou and Miyu settled in the Emiya residence. Kiritsugu was researching on how to use Miyu before she loses her ability and treated Miyu coldly like a tool. However Miyu was attached to Shirou which caused Shirou to be emotionally conflicted.
After Kiritsugu's death, Shirou couldn't treat her like a tool or even raise her like a normal child, so she was confined within the Emiya residence. Shirou gave her textbooks for her education, but it was knowledge that didn't hurt her and for non-practical use. When Miyu became ten years old, she had developed more human expressions. When Shirou reminisces Kiritsugu's last moments, Miyu unknowingly activated her wish that Shirou and Miyu become true siblings. Shirou at that moment decided to treat her more like a sister and to take her outside to see the ocean. Shirou took Miyu back to the Sakatsuki residence and tried to explain to her about her family and his intentions on saving her. Shirou's explanation was interrupted by Julian, when he accused Shirou to be a thief. Shirou tried to tell Miyu to run but she didn't know where to run and Julian kidnaps her.
After Shirou won the Fifth Ainsworth Holy Grail War, she is teleported to another Parallel world through Shirou's wish upon the Holy Grail. She was taken in by Luvia in her search to rival Illya and Rin's partnership.
She takes her duties as a Magical Girl very seriously and is initially dismissive of Illya's abilities, but gradually warms up to Illya. She comes to value Illya's friendship greatly and is highly protective of her. She shows another side of her personality only when she is around her brother, where she behaves like the young girl she actually is.
Fate/kaleid liner PRISMA☆ILLYA
Miyu takes on the Edelfelt name after being taken in by Luvia. She lives in Luvia's mansion and acts as her maid to earn her keep. She uses Ruby's sister Kaleidostick, Sapphire, in order to transform, and she initially has a much better understanding of her abilities than Illya. She takes her role very seriously and treats Illya as a bothersome rival, though they eventually become close friends. She later gains the ability to use the Saber class card to embody the Heroic Spirit after seeing Illya perform it.
Fate/kaleid liner PRISMA☆ILLYA 2wei!
Miyu continues to fight as a Magical Girl, dealing with the mysterious 8th Class Card. After dealing with Illya's split personality, Kuro, Miyu and co. fight the Enforcer, Bazett. They defeated Bazett and cooperated with her to capture the 8th Card.
The 8th Card was revealed to have come from another Holy Grail ritual, and it gains a physical body, child Gilgamesh. Later it was revealed by Gilgamesh that she comes from a parallel world and she is a princess. After being saved from black Gilgamesh by Illya, Miyu was captured by Angelica and Beatrice from another world.
Fate/kaleid liner PRISMA☆ILLYA 3rei!!
Captured, Miyu is held captive in Ainsworth's stronghold.
Later it was revealed her original name was Miyu Sakatsuki. She was a "Child of God", a child from an occult religion that worships children below 10 years old as gods, and this granted her wish-granting power, not unlike the Holy Grail. Kiritsugu and Shirou found her in an incident that brought her hometown into enormous black fire, and witnessed Miyu dispel the fire. Kiritsugu took her, wishing to utilize her wish-granting power to save the world.
Kiritsugu asked Shirou to use Miyu to grant his wish of world salvation, but Shirou couldn't do it because Miyu became more human each day. Kiritsugu died some years later, and Shirou continued to taking care of Miyu. At this time, Shirou and Miyu became really close. Miyu then used her power to wish for her and Shirou to become real siblings, in the anime this is seen by the change of eye color from her previous pink-gold colour to the same coloured amber eyes as Shirou.
Magical Girl Travel Diaries ~Prisma Corps~
She was captured by First Lady.
As a Magical Girl, Miyu is able to utilize the Mystic Code Kaleidostick, Sapphire, to tap into mana from parallel worlds and gains virtually unlimited mana. She also can utilize Class Cards; Including them to provide her Heroic Spirit's Noble Phantasm and Installing them to gains Heroic Spirit's abilities.
As the Child of God, she can grant wishes. However, this power seems to have disappeared after she turned 10 years old.