Celta have never won the league title nor Copa del Rey, although they have reached the final three times in the latter. One of the team's best seasons was 1970–71, when they finished unbeaten at home and were known as the "giant-killers". Celta came sixth that season and qualified for the UEFA Cup for the first time. The club finished in their best-ever position of fourth in 2002–03, qualifying for the 2003–04 UEFA Champions League, where they were eliminated by Arsenal in the Round of 16.
R.C. Celta de Vigo was formed as a result of the ambition of Vigo's teams to achieve more at national level, where the Basque sides had been their bête noire in the Spanish Championship. The idea was to merge both teams to create a more powerful team at national level. The standard-bearer of this movement was Manuel de Castro, known as "Handicap", a sports writer for the Faro de Vigo who, from 1915, began to write in his articles about the need for a unitarian movement. The slogan of his movement was "Todo por y para Vigo" ("All for and to Vigo"), which eventually found support among the managers of Real Vigo Sporting and Real Club Fortuna de Vigo. It was backed unanimously when de Castro himself presented the motion at the assembly of the Royal Spanish Football Federation in Madrid on 22 June 1923.
On 12 July 1923, at the annual general meetings (AGMs) of Vigo and Fortuna held at the Odeon Theatre and in the Hotel Moderno, respectively, the merger was approved. Thus the "Team of Galicia" was born, as it was dubbed. In the last AGM of Fortuna and Vigo to approve the formation of a new club held on 10 August 1923, the members decided upon the team's name. Various names suggested include "Real Unión de Vigo", "Club Galicia", "Real Atlántic", "Breogán" and "Real Club Olimpico". The latter name was popular, but they eventually decided on "Real Club Celta", an ethnic race linked to Galicia (see Celts). The first president of Celta was Manuel Bárcena de Andrés, the Count of Torre Cedeira. At this AGM, the squad was also decided, which numbered 64 players in total and included some notable players from both Fortuna and Vigo:
Goalkeepers: Isidro, Lilo and Rubido
Defenders: Otero, Pasarín, Juanito Clemente, Daniel y Kaíto
Celta had a dramatic reversal of fortune in 2003–04. In the previous season, they finished fourth in the league, putting them in the third qualifying round of the Champions League. Celta entered the group phase after eliminating Slavia Prague, and eventually reached the last 16 before being knocked out by Arsenal. However, their domestic form was disastrous, finishing second-to-last in La Liga, thus sealing their relegation to the Segunda División. Although the squad was heavily dismantled following the demotion, Celta earned an immediate return to the top flight after finishing second in 2004–05.
In the 2005–06 season, they finished sixth earning a return once more to the UEFA Cup. They made it to the last 16 in that competition as well before losing to German side Werder Bremen. The next year, 2006–07, Celta finished in 18th and were once again relegated to the Segunda División. At the end of June 2007, Celta avoided going into administration. However, if an agreement was not put in place between the club and its creditors within three months, then courts would declare the liquidation of the club's assets.
Due to heavy debt, the club was forced to sell many players and make tremendous cuts in the club's finances. Since then, they have been relying mainly on the reserve team, combined with some inexpensive signings. During the first three seasons in the Segunda División, the club struggled to avoid further relegation, all amid fears of the club's complete disappearance. This was a period of high instability, with constant changes of managers and players. In the 2010–11 season, however, the signings of striker David Rodríguez, winger Enrique de Lucas and manager Paco Herrera turned the situation around. The club finished sixth after a fantastic season and qualified for promotion. Nevertheless, they were eliminated in the first knockout round by Granada after a penalty shootout, the game having finished 1–1 in 90 minutes.
On 3 June 2012, Celta returned to La Liga after a five-year absence. In their first season back, they avoided relegation to the Segunda División on the final day after beating RCD Espanyol 1–0 to ensure a 17th-place finish. On 8 June 2013, Celta announced they had signed former Roma and then-Barcelona B manager Luis Enrique to lead the club for the 2013–14 season. Under Luis Enrique, Celta flourished, finishing ninth. After Luis Enrique's departure, his replacement, Eduardo Berizzo, led the team to eighth in La Liga during 2014–15, and the following season season saw Celta's highest finish in ten years, finishing in sixth position and earning a place in the 2016–17 UEFA Europa League
Celta's original team strip consisted of a red shirt, black shorts and blue socks. This was later changed at an unknown date to the traditional sky blue and white strip, representative of the Galician flag.
Celta had the longest-running sponsorship deal in Spanish football, and one of the longest-running in the world, with the French automobile manufacturer Citroën from 1985 to 2016. The company established its plant within walking distance from Balaídos in 1958, and first sponsored the club's women's basketball team in 1980. In 2016, the sponsor was changed to the Galician brewery Estrella Galicia, which had advertised on the back of the shirts since 2011. Their business deal with kit supplier, Umbro, was also one of the longest-running ones, from 1986 to 2010.
Like many other Galician clubs, such as Compostela and Racing Ferrol, the club badge is based on the red cross of Saint James. On top of the cross sits a sky blue shield with two letter "Cs" (Club Celta). In 1923, Celta became one of several Spanish football clubs that were granted patronage by the Spanish crown and thus entitled to use Real (Royal) in their names and the royal crown on their badge. This right was granted to Celta by Alfonso XIII, and the club subsequently became known as Real Club Celta de Vigo. During the Spanish Second Republic (1931–1936), the title Real was removed from the club's name and the royal crown was taken off the club crest; however, it was returned under the Spanish State.