She graduated from the classical gymnasium and high school of music, where she perfected the cello. As she finished music school and played the cello, she wanted to continue with the music academy, but in 1985, came the role of Laura Lenbach, in Miroslav Krleža's "In Agony",and she graduated in acting at the Academy of Dramatic Art in Zagreb, in the class of Professor Tomislav Durbešić.
She was born on March 25, 1963 in Zagreb. She was born to her mother, Marija, a Croatian language teacher, and her father, Fabijan Šovagović, one of the greatest Croatian actors, who with his work and contribution greatly marked and raised the quality of an era of acting in Croatia to another level. Anja's younger brother, Filip, is also involved in acting and directing.
As a seventeen-year-old girl, she collaborated with Dragan Despot, whom she married six years later. In marriage, they had a son, Jakov, and a daughter, Ana.
As she performed as an actress in the theater, she could not imagine herself in a film role. For a while, she refused film offers, until she accepted the role of Stella (in 1990), a troubled delinquent who escapes from a Zagreb, thereby making a turning point in her career.
After “Stela”, Anja shot a dozen films such as "Baka Bijela", a film from 1992, then the film "Sisters", also from 1992, then Bruno Gamulina's film, "Polagana predaja" from 2001, based on the novel of the same name by Goran Tribuson. The film was declared the best film, won five Golden Arenas.
She collaborated with numerous theaters in Croatia, as well as in the drama program of the Croatian Radio. In her career, she played over forty leading roles in the theater and thus entered the ranks of the greats of Croatian theatre.
In addition to achieving great success as a musician, and above all as an actress, Anja also loved literature and writing. She made her debut with the literary collection of theatrical essays "Wild Freedom". She also published the autobiographical book "When you have something to give", which she dedicated to her childhood.
She is the winner of many theater and film recognitions and awards. It is worth noting that she was awarded the Order of Danica Croatia. In 1987, she received the award for the best young actress at Gavella's evenings. She received the Golden Smile award for the best leading role at the Days of Satire for the role of Natalija Petrovna in Goldoni's "Summer Trilogy". She is also the bearer of the Memorial of Homeland Gratitude.