Sverak was raised in the Czech film world as the son of star actor Zdenek Sverak, who also starred in and scripted "Kolya". Graduating from the Film Academy of Music and Drama (FAMU) in Prague, he directed his first short film, "Good-Bye Little Station" in 1984. (Sverak was rejected in his application for the narrative branch of the school and had to take courses in the documentary branch instead.) While still a student, he began directing for Czech TV with "Conversation" (1986). His graduation film from FAMU, "Oil Gobblers" (1988), won a special student Academy Award. and he garnered particular attention with the 1986 short "Space Odyssey II", which told the story of two retired women living on a snow-bound estate.
Sverak made his feature film debut with "Elementary School", the story of a new teacher in post-war Prague. "Accumulator I" (1994), a science fiction parody about a man overcome by lethargy after watching too much TV, was awarded the Czech Lion Prize by the Czech Film and Television Academy as most popular film of 1994, and also won the Critic's Week Prize and the Media Prize at the Venice Film Festival. The director then steered a new course to "The Ride" (1994), a low-budget ($30,000) and off-beat road picture revolving around two men, a girl and her jealous boyfriend. The more ambitious "Kolya" followed; it recounted the story of a rascally bachelor talked into marrying a Russian woman so she can remain in Czechoslovakia and then finds himself raising the woman's six-year old son when she runs off. Set against the so-called Velvet Revolution of 1989, "Kolya" earned the Golden Globe Award and an Oscar nomination for Best Foreign-Language Film.