The name of Rapid București comes not from football but from the railway, chosen by the club's founders in honour of the fast passenger locomotives operated by their employer, the Romanian State Railways. The club was established on 25 June 1923 by a group of workers at the Grivița railway workshops in Bucharest, who formally registered it as the "CFR Cultural and Sporting Association", with Teofil Copaci as its first president. Its earliest years were precarious: in 1925 the young club was even suspended from the Bucharest championship for repeatedly failing to field a full team, often for lack of proper equipment.
Rapid's first golden age began in the 1930s, when, despite never winning the league title before the war, the club dominated the Cupa României, claiming the inaugural edition in 1933-34 and then reeling off six consecutive triumphs between 1936-37 and 1941-42. Star forward Giussy "the Blond Wonder" Barátky, who represented both Hungary and Romania internationally, led the line in June 1940 when Rapid beat Hungária MTK in Budapest to become the first Romanian club ever to win there, part of a run to the final of the 1940 Mitropa Cup that was never played because of the outbreak of war.
Under communism, with the state's resources funnelled into the newly created army and secret police clubs, Steaua and Dinamo, Rapid's railway-worker roots left it as Bucharest's "third club", though the team's terraces became known for a rebellious streak that at times spilled into open anti-regime sentiment. Sporting success eventually followed, with Rapid winning its first national championship in 1966-67, and adding further titles in 1998-99 and 2002-03. In European competition, the club's best runs came in the 1973 Cup Winners' Cup and the 2005-06 UEFA Cup, both ending in the quarter-finals. In total, Rapid has won three Romanian league titles, a record 13 Cupa României trophies and four Romanian Super Cups.
Financial collapse led to the club's bankruptcy in 2016, but Rapid was refounded and returned to Liga I in 2021. Nicknamed Feroviarii ("The Railwaymen") and playing in white and burgundy, Rapid continue to represent the Giulești district at the Superbet Arena – Giulești, a modern stadium built on the site of the club's historic ground and popularly still known simply as Giulești.