His first novel, written at the age of 18 after having finished high school, was Hüvasti, kollane kass (Goodbye, Yellow Cat).

[1] Unt was married to the television journalist and screenwriter Ela Tomson from 1965, until their divorce in 1973.

In 1980, he was named an Honored Writer of the Estonian SSR, and that same year he became one of the signatories to the Letter of 40 intellectuals.

Four successive novels, Võlg (The Debt, 1964), Elu võimalikkusest kosmoses (On the Possibility of Life in Space, 1967), Kuu nagu kustuv päike (The Moon like the Outgoing Sun, 1971), and Must mootorrattur (The Black Motorcyclist, 1976), established his reputation as a major writer.

Several of his novels have been adapted for film since his death, including Sügisball ("Autumn Ball") in 2007 by Veiko Õunpuu.