Avi Lebovich was born April 2, 1972 in Yehud, a town east of Tel Aviv. He started on piano as a child and moved to trombone under Ray Parnes, the Israel Philharmonic Orchestra's principal trombonist and the teacher behind most of the country's serious trombone playing, then went on to the Thelma Yellin High School of the Arts.

In 1992 Lebovich left for Manhattan, studying at The New School and later at Mannes College of Music. Two other Israelis made the same move that year, bassists Avishai Cohen and Omer Avital, and the three of them are routinely described in Israeli jazz histories as the generation that proved the trip was possible. Lebovich's own break came from a phone call. He cold-called Slide Hampton, told him he played at least as well as Hampton did, and got a meeting; Hampton listened, took him on, and put him in his band. Two years later Hampton brought him to Lincoln Center for the JVC Jazz Festival with Slide Hampton & the Jazz Masters. Their working relationship produced Lebovich's 1996 debut, Shades of Brass, cut for Pierre Sprey's audiophile label Mapleshade under the billing Avi Lebo Double Trombone Quintet: two trombones, Lebovich and Hampton, with Larry Willis producing and playing piano, Steve Novosel on bass, and Jimmy Cobb on drums.

Mid-decade he was invited to London to audition for Incognito, the flagship band of the British acid-jazz wave, and stayed two years, appearing on Beneath the Surface (1996), Blue Moods (1997), and the live Last Night in Tokyo (1997), all on Talkin' Loud. A tune he wrote in London for Incognito bandmate Chris Demargary, "Mr. Demargary," was picked up for The New Real Book and recorded by Jason Lindner's Ensemble on Premonition (2000). Back in New York he worked across an unusually wide range of bandstands: James Moody's Young at Heart (Warner Bros., 1996), Bootsy Collins's Fresh Outta 'P' University (1997), Philip Bailey's Steppin' Through Time (1997), two Ben Wolfe records for Mons, and Avishai Cohen's Stretch Records albums Colors (2000), on which Lebovich also contributed a composition, and Unity (2001) with the International Vamp Band. He has also cited work with Chick Corea, Milt Jackson, Roy Hargrove, Brad Mehldau, and Larry Willis from those years, though not all of it reached record. His second album as a leader, Constant Chase, was recorded at Water Music in Hoboken in 2001 and released on Fresh Sound New Talent in 2002, with Avishai Cohen on bass, Antonio Sanchez on drums, Alex Norris on trumpet, Luis Perdomo on piano, and Claudia Acuña singing.

Lebovich came home in 2003 and immediately built the thing he is best known for: a 12-piece jazz ensemble he called simply The Orchestra, billed in English as Avi Lebovich & the Orchestra and in Hebrew as תזמורת הג'אז הישראלית, the Israel Jazz Orchestra. It grew out of a quintet with Yair Slutzki, Amit Friedman, Itamar Gross, Micky Warshai, and Ron Almog, and it kept a big-band front line while writing for it as though the rhythm section came from somewhere else entirely: electronic textures, Latin and Middle Eastern rhythmic material, funk and rock backbeats, and a Radiohead cover ("There There") sitting next to Oliver Nelson's "Stolen Moments." The debut, Groove Collage, appeared on Hed-Arzi in August 2008 with Amos Hoffman on oud and Oz Noy on guitar. Volcano followed in February 2013, self-released, with Yonatan Albalak on guitar, Nitai Hershkovits guesting on keyboards and piano, cellist Maya Belsitzman, and Hadag Nahash frontman Shaanan Streett singing on "Horses." Then came a Yaakov "Yankale" Rotblit songbook album drawn from a 2011 benefit concert, a full-length collaboration with composer and flutist Shem-Tov Levi, בחברה טובה (Bechevra Tova, "In Good Company," Kame'a, 2015), and Nine Tales One Story (2018), which brought in trumpeter Avishai Cohen and Hershkovits as guests. The band has toured Canada, France, Italy, Germany, and South Africa, and its artistic direction has since passed to Almog and Slutzki, two of its own players.

The other half of his career is arranging and record production for singers. He composed, arranged, and produced Shaanan Streett's first solo album, הבזק אור חולף (2007), and the duet he wrote for it brought him to the attention of Chava Alberstein, who handed him a Nadav Levitan lyric as a test. The result was more than a year of sessions and her 45th studio album, טבע האדם (The Nature of Man, NMC, April 2008), for which Lebovich wrote the music, the arrangements, and the production while Alberstein, unusually, sang and did nothing else. The same year he arranged most of Din Din Aviv's חופשייה בין עולמות. As a writer for large ensembles he has supplied charts to Michel Camilo's big band, Chico O'Farrill's Afro-Cuban big band, the Smalls big band, and Slide Hampton's orchestra. He won the Prime Minister's Prize for Composers in 2005 and the Landau Prize for the Performing Arts for 2006/7 in the jazz composition and arranging category, and he teaches jazz trombone in the jazz department of the Israeli Conservatory of Music in Tel Aviv.

Lebovich plays with what Mapleshade's own liner notes called a super-warm, mellow tone, set deliberately against Slide Hampton's harder attack on their twin-trombone debut, and Hampton remains the acknowledged model behind his playing and his writing for brass. What separates him from the other Israeli players who went to New York in 1992 is that he came back and put his energy into a book rather than a horn: the Orchestra's repertoire is arranger's music, built on a conventional big-band section layout that gets fed drum-and-bass, Latin percussion, oud, electronics, and rock covers, with the ethnic material treated as rhythm and color rather than as a costume. That same arranger's instinct is what Israeli pop singers hire him for, and it produced the most discussed instance of it when Chava Alberstein, a songwriter for four decades, gave up composing and playing on an entire album so that his harmonic language could take over. Israeli critics and the Landau and Prime Minister's prize committees have consistently classed him as a composer and arranger first, trombonist second.