Influenced by music from a young age by her record store owner father, Costa became one of Brazil's foremost female Tropicalismo movement singers and guitar players during the late 1960s and 70s. Along with her friend Maria Bethânia, she became a political activist often coming into conflict with Brazil's government when dissent in music was censored. Fellow musician Caetano Veloso, Maria Bethania's brother, introduced her to Gilberto Gil and Tom Zé in 1963, who managed to secure her a recording deal in São Paulo for her debut album Domingo with Caetano Veloso. In 1973, the cover of Costa's album India was censored due to her wearing a risque red bikini. Costa has recorded songs composed by a number of Brazil's most popular songwriters such as Tom Jobim, Jorge Ben Jor, and Erasmo Carlos. In 1982 the single "Festa Do Interior" from the double album Fantasia became her biggest ever hit, going multi-platinum by the end of the year. Costa appeared in the 1995 film The Mandarin (Mandarim, O) as the singer Carmen Miranda. She has recorded songs in Portuguese, Spanish and English.