Anders Eliasson was born in 1947 in Borlänge. Even in his early years he showed a strong interest in music. He soon took up playing the trumpet, joining various jazz bands in his home town.
In 1966 Eliasson was accepted as a student at the Stockholm academy of music. Ingvar Lidholm became his teacher in composition and Ligeti and Ib Nörholm may be mentioned as other teachers who were to be important to Eliasson's musical development.

At the beginning of the 1970s Anders Eliasson wrote a number of momentous musical works.They were characterised by a striving to achieve a comprehensive perspective but at the same time they were marked by a strident poetic style; superfluousness was rejected.
By the end of the 70s Eliasson wrote his significant piece, Canto del vagabondo, thereby displaying a new direction in his artistic and musical forms of expression. Canto del vagabondo is based on Carl von Linné's Laplandish Journey and the work may be described as a consummate composition. It was written for boy soprano, female choir, and orchestra and in 1983 it rendered Eliasson the great Christ Johnson Award.

Anders Eliasson has also created many intriguing pieces for chamber music, for instance Quintetto per clavicembalo e quartetto (1985) and Quartetto d'archi (1991). (Almost all of Eliasson's musical works go by Italian titles.)
In the 1980s Eliasson wrote his first symphony which was soon to be followed by others, for instance Symphony no 3 for saxophone (1989).