Born in Fair Lawn, New Jersey, United States,^[4] Swallow studied piano and trumpet, as a child, before turning to the double bass^[2] at age 14. While attending a prep school, he began trying his hand in jazz improvisation. In 1960, he left Yale University, where he was studying composition, and settled in New York City, playing at the time in Jimmy Giuffre's trio along with Paul Bley.^[1] After joining Art Farmer's quartet in 1963,^[1] Swallow began to write. It is in the 1960s that his long-term association with Gary Burton's various bands began.^[1]

In the early 1970s, Swallow switched exclusively to electric bass guitar, of which he prefers the five-string variety. He was first introduced to the electric bass while doing a music trade show with Gary Burton and trying a Gibson EB-2.^[5] Along with Monk Montgomery and Bob Cranshaw, Swallow was among the first jazz bassists to do so (with much encouragement from Roy Haynes, one of Swallow's favorite drummers). He plays with a pick (made of copper by Hotlicks), and his style involves intricate solos in the upper register. He was one of the early adopters of the high C string on a bass guitar.

From 1974 to 1976, Swallow taught at the Berklee College of Music. He contributed several of his compositions to the Berklee students who assembled the first edition of The Real Book. He later recorded an album, Real Book, with the picture of a well-worn, coffee-stained book on the cover.

In 1978, Swallow became a member of Carla Bley's band.^[1] He was Bley's romantic partner from the 1980s until her death in 2023. He toured extensively with John Scofield in the early 1980s, and has returned to this collaboration several times over the years.

Swallow has consistently won the electric bass category in DownBeat yearly polls, both Critics' and Readers', since the mid-1980s. His compositions have been covered by, among others, Jim Hall (who recorded his very first tune, "Eiderdown"), Bill Evans, Chick Corea, Stan Getz and Gary Burton.

As leader/co-leader edit

• Hotel Hello with Gary Burton (ECM, 1974)
• Home – music to poems by Robert Creeley (ECM, 1980)
• Night-Glo with Carla Bley (Watt, 1985)
• Carla (Xtra Watt, 1987)
• Duets with Carla Bley (Watt, 1988)
• The Life of a Trio: Saturday with Paul Bley and Jimmy Giuffre (Owl, 1989)
• The Life of a Trio: Sunday with Paul Bley and Jimmy Giuffre (Owl, 1989)
• Swallow (Xtra Watt, 1991)
• Go Together with Carla Bley (Watt, 1992)
• Real Book (Xtra Watt, 1993)
• Songs with Legs (Watt, 1994) with Carla Bley and Andy Sheppard
• Parlance (Instant Present, 1995) with John Taylor
• Deconstructed (Xtra Watt, 1996)
• Are We There Yet? with Carla Bley (Watt, 1998)
• Always Pack Your Uniform on Top (Xtra Watt/ECM, 2000)
• Noisy Old Men (Jam, 2002) with Mick Goodrick, John Abercrombie, Gary Chaffee
• Damaged in Transit with Chris Potter and Adam Nussbaum (Xtra Watt, 2003)
• So There with Robert Creeley (Xtra Watt, 2006)
• L'Histoire du Clochard with Ohad Talmor (Palmetto, 2007)
• Carla's Christmas Carols with Carla Bley and the Partyka Brass Quintet (Watt, 2009)
• Playing in Traffic with Ohad Talmor and Adam Nussbaum (Auand, 2009)
• Into the Woodwork (Xtra Watt, 2013) – recorded in 2011
• The New Standard with Jamie Saft and Bobby Previte (RareNoise, 2014)
• Loneliness Road with Jamie Saft, Bobby Previte and Iggy Pop (RareNoise, 2017)

As sideman edit

• Watt/XtraWatt – the recording company run by Swallow and Bley
• Steve Swallow and Carla Bley video interview about Dreams So Real and working with ECM Records