Creation and Evolution
The Red Room was created by Devin Grayson and J.G. Jones. It first appeared in Black Widow #1.
As time progressed, the Red Room attempted to create another brainwashing and training program, the Wolf Spider Ops. The goal in this was to train young boys. However, after starting with only one member, it ultimately failed.
The Red Room is most famous for training one of Marvel's greatest heroes, Black Widow. She was taken to the Red Room at a young age along with 28 other orphan girls and was trained in advanced combat and trained to be a spy. She, along with the other girls, were brainwashed and manipulated by the Red Room. They physically enhanced Natasha while she was there and set her up to become an agent of the KGB, before she ultimately betrayed them and became a hero.
After Black Widow betrayed them, the Red Room started training another young girl, Yelena Belova, to be the next Black Widow. They succeeded in training her and setting up a competition between her and Natasha, but ultimately failed when Natasha helped Yelena betray them as well.
Nadia van Dyne
Nadia van Dyne was, like Romanoff and Belova, enrolled into the Red Room at a young age. Instead of using her to train her, though, the Red Room set her up to do work for them via science. However, Nadia ended up escaping and, like the women before her, betrayed them.
While not having been trained by the Red Room, the Winter Soldier has a notable connection to them. He trained and eventually fell in love with one of their members, Natasha.
In Secret Empire, Black Widow was the leader of the Red Room. Members of this version of the group included Hulk, Spider-Man, Viv, Falcon, Ironheart, and Wasp. Wasp almost ended up leaving the Red Room when she recognized Black Widow's manipulation tactics.
In this version, Black Widow and the Champions founded the Red Room with the intention of eliminating Captain America.
Marvel Cinematic Universe
In the Marvel Cinematic Universe, the Red Room, much like its comics' counterpart, trained Black Widow and other young girls such as Yelena Belova and Melina Vostokoff. It was led by Dreykov and most notably protected by his daughter, Taskmaster. In the MCU, the girls were not given physical enhancements, but were still brainwashed and manipulated. After Black Widow had betrayed the Red Room, she and Hawkeye, with S.H.I.E.L.D.'s help, went on a mission in Budapest to take down the Red Room. They believed they had succeeded but it was later revealed in the Black widow film that the Red Room was still active. Natasha allied with Yelena, Melina, and Red Guardian in order to finally take the Red Room down and free the Widows.
Marvel Avengers Academy
In Marvel Avengers Academy, the Red Room trained Black Widow in ballet.
In Marvel's Avengers, the Red Room trained Black Widow similarly to how they did in the comics. There was also a Red Room Takeover event where players played as a hero from their roster to go up against Yelena Belova, who had taken over the HARM Room.
Marvel Future Revolution
In Marvel Future Revolution, the Red Room trained Black Widow similarly to how they did in the comics.