Franz Leopold Maria Möst was born in Linz, Austria, and later studied under the composer Balduin Sulzer. As a youth in Linz, he studied the violin and had developed an interest in conducting. After suffering injuries in a car crash that led to nerve damage, he stopped his violin studies and shifted full-time to conducting studies.
In 1985, Möst assumed the stage name Welser-Möst on suggestion of his mentor, Baron Andreas von Bennigsen of Liechtenstein, thus paying homage to the city of Wels where he grew up. In 1986, he was adopted by Bennigsen. In 1992, Welser-Möst married Bennigsen's former wife, Angelika. His first major debuts were at the Salzburg Festival in 1985, followed by the London Philharmonic Orchestra in 1986 and the Orchester Musikkollegium Winterthur in 1988.
Between 1986 and 1991 Welser-Möst served as the principal conductor of Norrköping Symphony Orchestra, Sweden, and in 1990 he became principal conductor of the London Philharmonic Orchestra (LPO). His LPO tenure was controversial, with London critics giving him the nickname "Frankly Worse than Most". He concluded his LPO tenure in 1996.
From 1995 to 2000, he was music director with the Zürich Opera House. He became general music director of the Zürich Opera in September 2005, with an original commitment to the Opera through 2011. However, he stood down from the Zürich post in July 2008, after having agreed to serve in the same capacity at the Vienna State Opera.
In the USA, Welser-Möst made his American conducting debut with the St. Louis Symphony in 1989. He first guest-conducted the Cleveland Orchestra in February 1993. Welser-Möst became music director of the Cleveland Orchestra with the 2002/2003 season, for an initial contract of five years. At the end of his first season, his contract was extended for another five years. In June 2008, the orchestra announced a further extension of his contract with the Cleveland Orchestra through the 2017/2018 season. In October 2014, the orchestra announced an additional extension of his Cleveland Orchestra contract through the 2021-2022 season. In September 2019, the Cleveland Orchestra announced the newest extension of Welser-Möst's contract through the 2026-2027 season.
Welser-Möst first conducted at the Vienna State Opera in 1987, as a substitute for Claudio Abbado in a production of Gioachino Rossini's L'italiana in Algeri. On June 6, 2007, the Austrian government announced the appointment of Welser-Möst as Generalmusikdirektor of the Vienna State Opera, effective September 2010, alongside Dominique Meyer as director (Staatsoperndirektor). In September 2014, he announced his resignation from the Vienna State Opera, effective immediately.
During his tenure with the LPO, Welser-Möst had established an exclusive recording contract with EMI. His 1996 recording of Franz Schmidt's Symphony No. 4 received the Gramophone Award for Best Orchestral Recording. The CDs of Anton Bruckner's Mass No. 3 and Te Deum and works of Erich Wolfgang Korngold both received Grammy Award nominations for "Best Classical Album." EMI struck a similar deal with Welser-Möst to record performances at the Zürich Opera and has released a number of DVDs of his Zürich opera productions. In 2008 EMI reissued many of Welser-Möst's earlier recordings in an eight CD set. In October 2007, Deutsche Grammophon released the first commercial recording featuring Welser-Möst with The Cleveland Orchestra: Beethoven's Symphony No. 9 including German bass René Pape among the soloists. This recording was soon followed by a disc of Richard Wagner Lieder performed by the orchestra and soloist Measha Brueggergosman. Several DVDs have been issued as well, including Bruckner's 7th and 8th symphonies, at Severance Hall, and the 5th and 4th at the St. Florian Monastery near Linz where Anton Bruckner is buried.