Nørby was born in Copenhagen, Denmark, the daughter of opera singer Einar Nørby. She studied two years at the Danish Royal Theatre. She has received a number of awards and recognitions including the Ingenio et Arti medal in 2006. At the 27th Guldbagge Awards she was nominated for the award for Best Actress for her role in Freud's Leaving Home.
She has been married a number of times. First in 1956 to architect Mogens Garth-Grüner, and then to actor Henrik Wiehe. She met her next husband, Italian pop singer/actor Dario Campeotto, on the set of a film they were shooting in 1962. They were married in 1963, and had a son, actor Giacomo Campeotto, in 1964. They were divorced in 1969. She married a fourth time to actor Jørgen Reenberg in April 1970. She is currently married to pianist/composer, Svend Skipper.