Gelman had been living in exile in Mexico for the past 20 years and his death was announced by the National Council for Culture and Art.
The cause of death was not immediately given. The newspaper Milenio, for which he wrote a weekly column, said Gelman died at home.
Gelman was considered one of the Spanish speaking world's greatest poets, and also stood out for his firm stand against impunity for military regimes that once ruled in Latin American countries including his native Argentina.
Gelman suffered the cruelty of the Argentine military regime that ruled from 1976 to 1983, as his son and daughter in law died during it.
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Besides poetry, Gelman worked as a journalist and translator. He published more than 30 books.
He was abroad, in Rome, when the military seized power in Argentina.
His odyssey of exile led him to Madrid, Nicaragua, New York and finally Mexico City.
His poetry was known for a sharp sense of humor, a touch of the absurd and defiance in the face of injustice.
His daughter in law Maria Claudia Garcia is still listed as missing. She was abducted in Buenos Aires in 1976 while pregnant and taken to neighboring Uruguay.
Her daughter was handed over the family of a Uruguayan police officer. The remains of the mother were never found.
Gelman fought for years and finally located his granddaughter in 2000. It was one of the most highly publicized cases of babies being abducted and given away by the regime during the Argentine military dictatorship.