Tras estudiar clarinete, saxo y flauta, se instala en California, a comienzo de los años 1940 y estudia con Shorty Rogers. Toca, ya profesionalmente, con Charlie Barnet (1947-1948), Art Mooney (1949) y, finalmente, en la big band de Stan Kenton (1950-1951). Después, colabora de forma continuada con los Lighthouse All Stars de Howard Rumsey, hasta 1956, convirtiéndose en uno de los principales representantes del West Coast jazz. Realiza varias giras por Europa y África, con Bob Cooper, introduciéndose más tarde en la fusión con la bossa nova, de la mano de Laurindo Almeida.

En los años 60 y 70, se dedica casi en exclusiva al trabajo de estudio y a la música para cine, aunque graba algunos discos, por ejemplo, con Ravi Shankar.^[1]​ Entre sus colaboraciones de estudio es particularmente célebre su solo de flauta para la canción California Dreamin' del cuarteto vocal The Mamas & the Papas. A partir de 1974, retoma su carrera como instrumentista, formando una banda (L.A.Four) con Almeida, Ray Brown y Shelly Manne. Algunos autores estiman que Shank fue el flautista más importante de la costa occidental, especialmente considerando sus exitosos dúos con el oboísta Bob Cooper.^[2]​ Sin embargo, a partir de los años 1980 abandona prácticamente este instrumento, en favor del saxo alto, en un estilo muy influenciado por Art Pepper y Zoot Sims, con un control perfecto de la sonoridad, delicado y con fraseo veloz.^[3]​

Falleció de una embolia, a los 82 años, en Tucson (Arizona), donde vivía, tras acabar de grabar un nuevo álbum.^[4]​

• 1954 The Compositions of Shorty Rogers (Nocturne)
• 1954 Bud Shank and Three Trombones (Pacific Jazz)
• 1955 The Saxophone Artistry of Bud Shank (Pacific Jazz)
• 1955 Bud Shank – Shorty Rogers – Bill Perkins (Pacific Jazz)
• 1955 Bud Shank and Bob Brookmeyer (Pacific Jazz)
• 1956 The Bud Shank Quartet (Pacific Jazz)
• 1956 Live at the Haig (Candid)
• 1956 Flute 'n Oboe (Pacific Jazz)
• 1956 Jazz at Cal-Tech (Pacific Jazz)
• 1956 Theme Music from "The James Dean Story" (World Pacific)
• 1956 Strings & Trombones (Pacific Jazz)
• 1957 Jazz Swings Broadway (Pacific Jazz)
• 1958 I'll Take Romance (World Pacific/Toshiba EMI)
• 1958 Misty Eyes (West Wind)
• 1958 The Swing's to TV (Pacific Jazz/Toshiba EMI)
• 1958 Bud Shank and Bill Perkins (Blue Note)
• 1958 Bud Shank in Africa (Pacific Jazz)
• 1959 Holiday in Brazil (World Pacific/Ubatuqui)
• 1959 Blowin' Country (World Pacific/Blue Note)
• 1959 Latin Contrasts (World Pacific)
• 1960 Bud Shank Plays Tenor (Pacific Jazz/Toshiba EMI)
• 1960 Slippery When Wet (World Pacific)
• 1960 Flute 'n Alto (World Pacific)
• 1961 Barefoot Adventure (Pacific Jazz)
• 1961 New Groove (Pacific Jazz)
• 1961 The Talents of Bud Shank (Kimberly)
• 1961 Swinging Broadway (Kimberly)
• 1962 Bossa Nova Jazz Samba (Pacific Jazz)
• 1962 Improvisations (EMI Angel/Angel)
• 1963 Brasamba! (Pacific Jazz)
• 1963 Folk 'n Flute (World Pacific)
• 1965 Bud Shank & His Brazilian Friends (Pacific Jazz)
• 1966 Bud Shank & the Sax Section (Pacific Jazz)
• 1966 Michelle (World Pacific/EMI Japan)
• 1966 Girl in Love (World Pacific)
• 1966 Flute, Oboe and Strings (World Pacific)
• 1966 California Dreamin' (EMI Japan)
• 1967 Brazil! Brazil! Brazil! (World Pacific)
• 1967 Bud Shank Plays Music from Today's Movies (World Pacific)
• 1967 A Spoonful of Jazz (World Pacific)
• 1968 Magical Mystery (World Pacific/EMI Japan)
• 1970 Let It Be (World Pacific)
• 1976 Sunshine Express (Concord Jazz)
• 1978 Heritage (Concord Jazz)
• 1979 Crystal Comments (Concord Jazz)
• 1980 Explorations (Concord Jazz)
• 1983 Yesterday, Today and Forever (Concord Jazz) with Shorty Rogers
• 1984 This Bud's for You... (Muse/32 Jazz)
• 1985 California Concert (Contemporary) with Shorty Rogers
• 1986 Serious Swingers (Contemporary)
• 1986 That Old Feeling (Contemporary)
• 1987 Bud Shank Quartet at Jazz Alley (Contemporary)
• 1989 Tomorrow's Rainbow (Contemporary)
• 1990 Tales of the Pilot: Bud Shank Plays the Music of David Peck (Capri)
• 1990 Lost in the Stars: Bud Shank and Lou Levy Play the Sinatra Songbook (Fresh Sound)
• 1990 Drifting Timelessly (Capri)
• 1991 The Doctor Is In (Candid)
• 1992 I Told You So (Candid)
• 1992 The Awakening (New Edition/ Vertriebsges.mb)
• 1995 New Gold (Candid)
• 1995 Lost Cathedral (Itm)
• 1996 Plays the Music of Bill Evans (Fresh Sound)
• 1996 Bud Shank Sextet Plays Harold Arlen (Jimco)
• 1997 Jazz in Hollywood (Original Jazz Classics)
• 1997 By Request: Bud Shank Meets the Rhythm Section (Muse/Milestone)
• 1998 A Flower Is a Lovesome Thing (Koch)
• 1999 After You Jeru (Fresh Sound)
• 2000 Silver Storm (Raw)
• 2002 Plays Tenor (Toshiba)
• 2002 On the Trail (Raw)
• 2003 Cool Fool (Fresh Sound)
• 2005 Bouncing with Bud and Phil (Capri)
• 2006 Taking the Long Way Home (Jazzed Media)
• 2006 Brazilliance, Vol. 1/Concert Creations for Guitar (FiveFour)
• 2007 Beyond the Red Door (Jazzed Media)
• 2009 Fascinating Rhythms (Jazzed Media)
• 2011 In Good Company (Capri)

Con Chet Baker

• Witch Doctor (Contemporary, 1953 [1985])
• The Trumpet Artistry of Chet Baker (Pacific Jazz, 1954)
• Chet Baker & Strings (Columbia, 1954)
• Chet Baker Sings and Plays (Pacific Jazz, 1955)
• Chet Baker Big Band (Pacific Jazz, 1956)

Con Elmer Bernstein

• The Man with the Golden Arm (Decca, 1956)

Con Buddy Bregman

• Swinging Kicks (Verve, 1957)

Con The Charlie Byrd Trio

• Brazilville (Concord Jazz, 1981)

Con Gene Clark

• Roadmaster (A&M Records, 1972)
• Firebyrd (Takoma, 1984)

Con Nat King Cole

• L-O-V-E (Capitol, 1965)

Con Buddy Collette

• Buddy Collette's Swinging Shepherds (EmArcy, 1958)
• At the Cinema! (Mercury, 1959)

Con Ron Elliott

• The Candlestickmaker (Warner Bros, 1970)

Con Maynard Ferguson

• Maynard Ferguson's Hollywood Party (EmArcy, 1954)
• Dimensions (EmArcy, 1955)
• Around the Horn with Maynard Ferguson (EmArcy, 1956)

Con Clare Fischer

• Extension (Pacific Jazz, 1963)

Con Jimmy Giuffre

• Jimmy Giuffre (Capitol, 1955)
• The Jimmy Giuffre Clarinet (Atlantic, 1956)
• Herb Ellis Meets Jimmy Giuffre (Verve, 1959) with Herb Ellis

Con Stan Kenton

• Innovations in Modern Music (Capitol, 1950)
• Stan Kenton Presents (Capitol, 1950)
• Popular Favorites by Stan Kenton (Capitol, 1953)
• This Modern World (Capitol, 1953)
• The Kenton Era (Capitol, 1940–54, [1955])
• Lush Interlude (Capitol, 1958)
• Stan Kenton Conducts the Los Angeles Neophonic Orchestra (Capitol, 1965)
• Hair (Capitol, 1969)

Con Barney Kessel

• Easy Like (Contemporary, 1953)

Con The Mamas & the Papas

• California Dreamin' (RCA Victor, 1965)

Con Shelly Manne

• The West Coast Sound (Contemporary, 1955)
• Manne–That's Gershwin! (Capitol, 1965)
• Daktari (Atlantic, 1967)

Con Sérgio Mendes

• Brasil '65 (Capitol, 1965)

Con Hugo Montenegro

• Colours of Love (RCA Victor, 1970)

Con Gerry Mulligan

• Gene Norman Presents the Original Gerry Mulligan Tentet and Quartet (GNP, 1953 [1997])
• I Want to Live (United Artists, 1958)

Con Oliver Nelson

• Skull Session (Flying Dutchman, 1975)

Con Harry Nilsson

• Duit on Mon Dei (RCA Victor, 1975)

Con Jack Nitzsche

• Heart Beat (Soundtrack) (Capitol, 1980)

Con Patti Page

• In the Land of Hi-Fi (EmArcy, 1956)

Con Shorty Rogers

• Cool and Crazy (RCA Victor, 1953)
• Shorty Rogers Courts the Count (RCA Victor, 1954)
• Collaboration (RCA Victor, 1954) with André Previn
• Martians Come Back! (Atlantic, 1955 [1956])
• Way Up There (Atlantic, 1955 [1957])
• Afro-Cuban Influence (RCA Victor, 1958)
• Chances Are It Swings (RCA Victor, 1958)
• The Wizard of Oz and Other Harold Arlen Songs (RCA Victor, 1959)
• Shorty Rogers Meets Tarzan (MGM, 1960)
• The Swingin' Nutcracker (RCA Victor, 1960)
• An Invisible Orchard (RCA Victor, 1961 [1997])
• The Fourth Dimension in Sound (Warner Bros., 1961)
• Bossa Nova (Reprise, 1962)
• Jazz Waltz (Reprise, 1962)

Con Pete Rugolo

• Introducing Pete Rugolo (Columbia, 1954)
• Adventures in Rhythm (Columbia, 1954)
• Rugolomania (Columbia, 1955)
• New Sounds by Pete Rugolo (Harmony, 1954–55, [1957])
• Out on a Limb (EmArcy, 1956)
• An Adventure in Sound: Reeds in Hi-Fi (Mercury, 1956 [1958])
• Rugolo Plays Kenton (EmArcy, 1958)
• The Music from Richard Diamond (EmArcy, 1959)
• Behind Brigitte Bardot (Warner Bros., 1960)
• The Original Music of Thriller (Time, 1961)
• 10 Saxophones and 2 Basses (Mercury, 1961)

Con Boz Scaggs

• Silk Degrees (Columbia, 1976)

Con Lalo Schifrin

• Music from Mission: Impossible (Dot, 1967)
• There's a Whole Lalo Schifrin Goin' On (Dot, 1968)
• More Mission: Impossible (Paramount, 1968)
• Mannix (Paramount, 1968)
• Bullitt (soundtrack) (Warner Bros., 1968)
• Che! (soundtrack) (Tetragrammaton, 1969)
• Kelly's Heroes (soundtrack) (MGM, 1970)

Con Ravi Shankar

• Improvisations (World Pacific, 1962)

Con Gábor Szabó y Bob Thiele

• Light My Fire (Impulse!-ABC, 1967)

Con Gerald Wilson

• Moment of Truth (Pacific Jazz, 1962)
• Portraits (Pacific Jazz, 1964)
• On Stage (Pacific Jazz, 1965)

Con Julie London

• All Through the Night: Julie London Sings the Choicest of Cole Porter (Liberty, 1965)

• Joachim E. Berendt: El Jazz: Origen y desarrollo (Fondo de Cultura Económica. Madrid, 1986. ISBN 84-375-0260-8)

• CARLES, Phillipe; CLERGEAT, André y COMOLLI, Jean-Louis: Dictionaire du jazz. Robert Laffont Edt., París, 1988. ISBN 2-221-04516-5

•   Wikimedia Commons alberga una categoría multimedia sobre Bud Shank.