François-Alexandre Galepides was born on February 14, 1929, in Paris, into a family of Greek descent, and became known by his stage name, Moustache, after his prominent moustache. In 1948 he joined the orchestra of the clarinetist Claude Luter, Les Lorientais, which played the cellar clubs of Saint-Germain-des-Prés, and with Luter he regularly accompanied Sidney Bechet in France. When Luter's band opened the Vieux Colombier club in 1949, Galepides was its drummer, alongside trumpeter Pierre Dervaux, trombonist Bernard Zacharias, pianist Christian Azzi and bassist Roland Bianchini. The Bechet-Luter session recorded in Paris on October 14, 1949, which produced "Les Oignons" and "Ridin' Easy Blues," lists him on drums. He is also on Luter's October 30, 1951 session with the clarinetist Mezz Mezzrow and, uncredited, on the Bechet concerts recorded at the Salle Pleyel on January 31 and March 12, 1952, issued by Blue Note as Jazz Festival Concert, Paris 1952.

From 1950 he led groups of his own. The Moustache Jazz Seven recorded for Pathé, with Guy Longnon on trumpet, Benny Vasseur on trombone, Hubert Fol on alto saxophone, Raymond Fol on piano and Roland Bianchini on bass, and its 10-inch LP Surprise Partie au Palm-Beach is dated 1953, followed by a second 10-inch LP in 1954. His Dixieland Jazz Band made a run of EPs and LPs for Véga from 1955, and in the mid-1950s he was the drummer on a Barclay EP by Lionel Hampton's quartet, Blues for the Hot-Club de Lille, with Michel Attenoux on soprano saxophone and Raymond Fol on piano. From 1956 he also recorded parody rock 'n' roll, first as Moustache et le Rock 'n' Roll on Véga and from 1958 on Barclay as Moustache et ses Moustachus, singing and playing drums, and including "Le Croque-Crâne-Creux" on a text by Boris Vian.

Alongside the music he built a second career as a comic and character actor, appearing in more than forty films from 1954, often as a musician: he is the drummer in Martin Ritt's Paris Blues (1961) and played Sergeant Garcia opposite Alain Delon in Duccio Tessari's Zorro (1975), a part he had already taken in six episodes of the children's television series Les Amis de Zorro in 1972. He also ran a restaurant that bore his name at 40 avenue Duquesne in Paris, managed the club Le Bilboquet in the 1960s, and from 1976 ran the jazz club of the Hôtel Méridien Étoile. A live album from that room, recorded on September 15 and 16, 1977, presents the saxophonist Michel Attenoux and vibraphonist Géo Daly; Moustache plays drums on two of its tracks. In 1967 he had formed Les Gros Minets with the pianist and singer Jean Constantin, Daly, trombonist Jordi Coll and Attenoux, a group that played at the Olympia and made singles for CBS from 1967 to 1969.

In 1978 he formed Les Petits Français with Attenoux, the clarinetist and tenor saxophonist Marcel Zanini and the trombonist François Guin, a band that turned popular French songs into jazz. Its best-known project came out of a friendship with the singer-songwriter Georges Brassens, whom Moustache had met at the Vieux Colombier club in Juan-les-Pins in the 1950s. Around 1976 he promised Brassens a set of his songs in jazz arrangements, and the result, Georges Brassens joue avec Moustache et les Petits Français (Philips, January 1979), put Brassens and his usual accompanists Pierre Nicolas and Joèl Favreau alongside Les Petits Français and five American guests then in Paris, the trumpeters Cat Anderson, Harry "Sweets" Edison and Joe Newman, tenor saxophonist Eddie "Lockjaw" Davis and pianist Dorothy Donegan, with arrangements by Attenoux. The double album won the Académie du disque français prize, the Grand Prix de la ville de Paris and Europe 1's Trophée no. 1 in 1979, and was reissued in 1980 as Brassens-Moustache jouent Brassens en jazz. After Brassens's death in 1981, Moustache recorded a second Brassens set in the summer of 1982 at Studio Le Vigilant in Nice, issued in 1983 as Hampton, Salvador, Clark Terry, Moustache et leurs amis jouent Brassens, with Lionel Hampton, the singer Henri Salvador and Clark Terry.

His last records were boogie-woogie and train blues albums with the pianist Jean Peron Garvanoff: Boogie Woogie Piano (Caravage, 1984), Blues for Mama Mous (Jazzmosphère, 1985) and Roule Train Blues (Sortie de Secours, 1986), the last with Moustache on drums and vocals. He also raced cars as a member of the Star Racing Team with other celebrities of the 1980s. On March 25, 1987, he was killed in a car accident at Arpajon, in the Essonne, when his car ran under a truck on the Route nationale 20 south of Paris. He was buried at the cemetery of Le Perreux-sur-Marne beside his wife, the dancer Simone van Lancker, whom he had married in Antibes in 1958 and who died in 1978.

Moustache belonged to the New Orleans revival that grew up around Claude Luter and Sidney Bechet in the Saint-Germain-des-Prés cellars after the war, and his drumming developed in that traditional, collective-improvisation setting. He was also a showman who sang, clowned and parodied, which made him a natural fit for the comic rock 'n' roll EPs of the 1950s, and he moved easily between that humor and straight Dixieland and swing. His later reputation rests largely on bringing jazz soloists to French song: the Brassens albums of 1979 and 1983 set the chanson repertoire for a front line of American swing-era players and French mainstream musicians, and the first of them was honored by several French prizes. Specific teachers or drummers he modeled himself on are not documented in the sources consulted. He worked as a restaurateur, club owner and actor throughout, so his output as a leader is modest and mostly French-label singles, EPs and LPs.