After finishing second in the French national youth football trials in 1949, Kopa began his professional career at age 17 with Angers in Ligue 2 and was transferred two years later to Reims, with whom he won French championships in 1953 and 1955. He won the 1953 Latin Cup with Reims, where they defeated Milan 3–0 in the final, and helped them reach the 1956 European Cup Final, which the team lost to Alfredo Di Stéfano’s Real Madrid, 4–3.

Kopa had first attracted attention in Spain when he played for France against Spain in a match in Madrid in March 1955, which led to the Spanish sports newspaper Marca nicknaming him “Little Napoleon”. Kopa was transferred to Real Madrid for the 1956–57 season, where he was soon joined by Ferenc Puskás. Despite playing as an inside right (Playing as the no. 10 is his usual position), Kopa’s time at Real Madrid was extremely successful as he helped the club to three successive European cup victories, being incredibly influential in the Spanish capital. Playing with the Spanish giants, Kopa won the Spanish league in 1957 and 1958. Kopa was also the first French player to win the European Cup when Madrid defeated Fiorentina 2–0 in the 1957 final. He would go on to be European champion again in 1958 and 1959, the last against former side Reims, where Just Fontaine played. In the 1959–60 season, Kopa returned to France to finish his career with Reims, where he won further Championnats in 1960 and 1962. In total, he scored 75 goals in 346 matches in France’s top flight, and was awarded the Ballon d’or by France Football in 1958.

With the France national team, Kopa scored 18 goals in 45 matches between 1952 and 1962. He played in the 1958 FIFA World Cup in Sweden, where he performed outstandingly and was one of the players of the tournament, scoring three goals as he led France to the semi-finals, where they succumbed to a strong Brazil team. The French team finished third in the tournament.

In March 2004, Kopa was named by Pelé as one of the top 125 greatest living footballers. In 2018, France Football’s Kopa Trophy, awarded to the best young football player in the calendar year was named in Raymond’s honor. The first recipient was fellow Frenchman Kylian Mbappé.

Kopa married Christiane, the sister of a team-mate of his at Angers. After retiring from the game he launched his own sportswear brand, eventually settling in Corsica. Kopa died in Angers on 3 March 2017, aged 85.