Korać was born in Sombor in the Kingdom of Yugoslavia. He started playing for BSK (OKK Belgrade) at the age of 16, and played as a left-handed forward-center. Popularly nicknamed "Žućko" ('Blondie'), he became one of the best, if not the best player of Yugoslavia in the 1960s. He was the best scorer of the Yugoslav league for seven seasons, a record.
He entered the Yugoslav national basketball team in 1958 and won five silver and one bronze medal with them, in a total of 157 international games.
Korać played professionally for OKK Beograd, reaching various European champions' competitions with the club. Korać was three times named as part of the best European selection. In a 1964/65 FIBA European Champions Cup first round return game against Swedish champions Alvik from Stockholm, he scored 99 points (the game ended 155-57).
Away from the basketball court, Korać enjoyed theatre, music and reading books. He was a senior undergraduate from the Faculty of Electrical Engineering (ETF) at the University of Belgrade. Surprisingly, he once turned down a contract offer to play with Red Star Belgrade.
On June 2, 1969, at the age of 30, Korać died in a car crash on a road near Sarajevo. In 1971, FIBA established Radivoj Korać Cup in his remembrance, and no basketball games were played in Yugoslavia on June 2nd again by decision of the country's Basketball Federation. The Europe-wide Cup folded in 2002. At that time, the basketball federation of Serbia-Montenegro renamed its national cup competition to Radivoj Korać Cup. He was enshrined in the FIBA Hall of Fame in 2007.