Yuriy Norshteyn was born in the village of Andreyevka, Penza Oblast, during his parents' World War II evacuation. He grew up in the Maryina Roshcha suburb of Moscow. After studying at an art school, Norshteyn initially found work at a furniture factory. Then he finished a two-year animation course and found employment at studio Soyuzmultfilm in 1961. The first film that he participated in as an animator was Who Said "Meow"? (1962).
After working as an animation artist in some fifty films, Norshteyn got the chance to direct his own. In 1968 he debuted with 25th October, the First Day, sharing directorial credit with Arkadiy Tyurin. The film used the artwork of 1920s-era Soviet artists Altman and Petrov-Vodkin.
The next film in which he had a major role was The Battle of Kerzhenets (1971), a co-production with Russian animation director Ivan Ivanov-Vano under whose direction Norshteyn had earlier worked on 1969's Times of the Year.
Throughout the 1970s Norshteyn continued to work as an animator in many films (a more complete list can be found at IMDB), and also directed several. As the decade progressed his animation style became ever more sophisticated, looking less like flat cut-outs and more like smoothly-moving paintings or sophisticated pencil sketches.
Norshteyn uses a special technique in his animation, involving multiple glass planes to give his animation a three-dimensional look. The camera is placed at the top looking down on a series of glass planes about a meter deep (one every 25-30cm). The individual glass planes can move horizontally as well as toward and away from the camera (to give the effect of a character moving closer or further away). He does not use computers in his work.
For many years he has collaborated with his wife, the artist Franchesca Yarbusova, and the cinematographer Aleksandr Zhukovskiy.
Scene from The Overcoat
Throughout the late 1970s and early 1980s, Norshteyn's animations were showered with both state and international awards. Then, in a bitter twist of irony, he was fired from Soyuzmultfilm in 1985 for working too slowly on his latest film, a (presumably) feature-length adaptation of Gogol's Overcoat. By that time he had been working on it with his usual small team of three people for two years and had finished ten minutes.
In April 1993, Norshteyn and three other leading animators (Fyodor Khitruk, Andrey Khryanovsky, and Edward Nazarov) founded the Animation School and Studio (SHAR Studio) in Russia. The Russian Cinema Committee is among the share-holders of the studio.
To this day, Norshteyn is still working on The Overcoat—his ardent perfectionism has earned him the nickname "The Golden Snail". The project has met numerous financial troubles and false starts, but Norshteyn has said that it currently has reliable funding from several sources, both from within and from without Russia. At least 25 minutes have been completed and more of the film is expected to be released at the end of 2007. A couple of short, low-resolution clips have been made available to the public:  and . The first 20 minutes of the film have also toured among various exhibits of Norshteyn's work in Russian museums. The full film is expected to be 65 minutes long.
Norshteyn wrote an essay for a book by Giannalberto Bendazzi about the pinscreen animator Alexandre Alexeïeff titled Alexeieff - Itinerary of a Master.
In 2005, he released a Russian-language book titled Snow on the Grass, featuring a number of lectures that he gave about the art of animation. That same year, he was invited as "guest animator" to work on Kihachiro Kawamoto's puppet-animated feature film, The Book of the Dead.