Veronica Swift was born Veronica O'Brien on May 14, 1994, in Charlottesville, Virginia, the only child of the bebop pianist Hod O'Brien and the singer Stephanie Nakasian. She took her professional surname from her father's birth mother, whose last name had been Swift. She grew up on the road with her parents and sat in at their gigs, and in November 2004, at the age of nine, she recorded Veronica's House of Jazz for her parents' HodStef label, with saxophonist Richie Cole alongside her father's rhythm section and her mother among the performers. Her second album, It's Great to Be Alive (2007), was made at thirteen, with O'Brien arranging and playing piano and Harry Allen guesting on tenor saxophone on seven tracks. She appeared in the Women in Jazz series at Dizzy's Club Coca-Cola at eleven and played trumpet in her school band.

She studied jazz voice at the University of Miami's Frost School of Music and graduated in 2016 with a bachelor's degree. While there she wrote a goth-rock opera, Vera Icon, about a homicidal nun, an outlet she has linked to her anger over her father's cancer. In 2015 she released Lonely Woman on HodStef, which includes two tracks with O'Brien at the piano and may hold his last recorded playing, and that year she took second place in the Thelonious Monk International Vocals Competition. The finals were held at the Dolby Theatre in Los Angeles; Jazzmeia Horn won and Vuyolwethu Sotashe placed third. Hod O'Brien passed away on November 20, 2016, at the age of 80.

After Miami she moved to New York City and sang on Saturday nights at Birdland, where she also worked with the Birdland Big Band. Let's Sail Away, an album credited to Jeff Rupert with Veronica Swift, was recorded on March 16 and 17, 2017, at Starke Lake Studio in Ocoee, Florida, and released that November. She sang on five tracks of pianist Benny Green's Then and Now (Sunnyside, 2018) and later on the title track of his Benny's Crib (2020), for which she also wrote the lyrics. She appeared with Chris Botti, Michael Feinstein and Wynton Marsalis and the Jazz at Lincoln Center Orchestra, with whom she sang "(Everybody's Waitin' For) The Man with the Bag" on Big Band Holidays II (Blue Engine, 2019).

Her Mack Avenue debut, Confessions (August 30, 2019), paired her with two trios: Emmet Cohen's on most tracks and Green's on three. It won her Best New Artist and Best Vocal Release in the JazzTimes Readers' Poll that year. This Bitter Earth (March 19, 2021) was recorded at Sear Sound in New York with Cohen, bassist Yasushi Nakamura and drummer Bryan Carter. It takes its title song from a 1960 Dinah Washington recording and sets it beside show tunes such as "How Lovely to Be a Woman" and "You've Got to Be Carefully Taught," Dave Frishberg's "The Sports Page," Bob Dorough's "You're the Dangerous Type," and a jazz reading of the Dresden Dolls' ballad "Sing."

Her third Mack Avenue release, Veronica Swift (September 15, 2023), is a deliberately wide-ranging record she calls "transgenre." It runs from "Do Nothing Till You Hear from Me" and the bossa nova "Chega de Saudade" through an operatic "Je Veux Vivre" to Queen's "Keep Yourself Alive" and "Don't Rain on My Parade," with Benny Benack III on trumpet on two tracks and Alexander Claffy on bass on two. Swift co-produced it with Brian Viglione, the drummer of the Dresden Dolls, and Mariano Aponte. Viglione had come across her through her cover of "Sing" in 2021 and joined her touring band in 2023 and 2024; in the fall of 2024 the two formed DAME, a Los Angeles rock band in which she sings and plays keyboard. DAME played its first show at Jazz in Marciac in France on July 26, 2025, and made its American debut at the Viper Room in Los Angeles on September 5, 2025. In 2024 she was also named a Chevalier of France's Order of Arts and Letters.

Jazz work has continued alongside the rock project. She sang the Christmas single "Let It Snow!" with Jeff Goldblum and the Mildred Snitzer Orchestra on Decca (2024), and in March 2026 she appeared on two tracks of the Javier Nero Jazz Orchestra's Alkebulan (Outside In Music). A 2026 album, Home, has been announced, with pianist Mathis Picard and bassist Philip Norris.

Swift came up inside the bebop tradition through her parents, and her teenage album It's Great to Be Alive already includes Charlie Parker's "Yardbird Suite" and "Confirmation." Her adult work keeps that vocabulary of fast tempos, scat and horn-like phrasing while moving freely between songbook standards, musical theater, bossa nova, opera and rock; she has said that only about a quarter of her repertoire comes from the Great American Songbook. The critic Bill Milkowski named her one of twelve new jazz artists to watch in 2019 and praised her perfect pitch and phrasing, and Jazzwise gave the 2023 album four stars, noting its Queen-style guitar and massed vocal harmonies. Her sidemen on the Mack Avenue records, among them Emmet Cohen, Benny Green and Benny Benack III, come from the swing-and-bebop mainstream of New York, and her choice of a Dinah Washington song as the title track of This Bitter Earth places her in a line of singers who treat pop and theater songs with jazz rhythm and harmony. Since 2024 she has used DAME for original hard-rock material, and her jazz and rock work now run in parallel.