Robert Anchipolovsky was born in 1974 in Kiev, in what was then Soviet Ukraine, into a family already built around music: his father was a jazz saxophonist and his grandfather taught at the Kiev Academy of Music. He began on clarinet at nine under their direction, then switched to alto saxophone at fourteen, in 1988, drawn specifically to the jazz he heard in Charlie Parker and John Coltrane rather than the classical repertoire his family had started him on. He immigrated to Israel in 1991 and immediately joined its jazz scene, then spent 1992 to 1996 in New York studying under Phil Woods, one of the defining alto saxophonists of the postwar era. Woods later said of his student, "Robert is one of the most talented alto saxophonists I have ever heard." Anchipolovsky returned to Israel in 1996 and has led his own bands there ever since.

In 2004, Anchipolovsky won first prize in the alto saxophone category of Italy's International Massimo Urbani Award ("I.M.U.A."), an award named for one of Italian jazz's own most celebrated, Parker-indebted altoists; the win is listed directly on the award's own official history page. His European ties run further than that one credential: his sole documented studio-adjacent recording is a duet with Phil Woods himself, "Star Eyes," archived on Woods's own official discography site with a European rhythm section (pianist Tony Pancella, bassist Dima Grodsky, drummer Pietro Iodichi). Four years after Woods's death in 2015, Anchipolovsky led a tribute concert at the Birdland Jazz Club in Neuburg an der Donau, Germany, a room Woods himself had played more than once, and wore the leather cap Woods had left him during the performance.

Since at least 2016, Anchipolovsky has built a recurring concert franchise on the Tel Aviv-area club circuit, leading a different single-legend tribute program every year or two: a 2016 quartet date at Tel Aviv's Shnabel Club worked through George Gershwin's songbook; a July 2022 concert billed as "Bird With Strings" paired him with a small string, oboe, and harp ensemble on Charlie Parker's own strings recordings; club dates in March 2023 (Rodeo Haifa) and July 2024 (Salon de Tango, Ramat Gan) were devoted to baritone saxophonist Gerry Mulligan's Cool Jazz songbook, with Anchipolovsky switching to baritone sax for the occasion; a January 2024 date at Country Goren Goldstein paired him with fellow saxophonist Shlomi Goldenberg for a tribute to Stan Getz; an August 2024 quintet date at Salon de Tango brought in London-based alto saxophonist Jamie O'Donnell for a program of Parker, Dizzy Gillespie, Thelonious Monk, Hank Mobley, Miles Davis, and Wayne Shorter; and a December 2024 quartet date with pianist Gregory Fain at Shablul worked through Michel Legrand, Vernon Duke, Quincy Jones, and Cole Porter. His regular working rhythm section across many of these dates has included pianist Ilya Perelstein and bassist Dima Grodsky, with drummers Roy Kimelman and Gideon Paskovich rotating through.

Per his own promotional biography, Anchipolovsky has performed at festivals across Israel, Europe, Russia, and the United States, including the Red Sea Jazz Festival (with a duo set alongside Erez Barnoy documented from 2002), the Tel Aviv Jazz Festival, the Paris International Jazz Festival, the Wichita Jazz Festival, and Russia's Baikal and Sochi jazz festivals, and has shared stages over the years with Randy Brecker, Curtis Fuller, Louis Hayes, Jimmy Cobb, Kenny Barron, Larry Willis, Steve Turre, Valery Ponomarev, Mulgrew Miller, Vincent Herring, Deborah Brown, Kevin Mahogany, and Adam Makowicz, plus European-circuit collaborators including Tony Pancella, Max Ionata, and fellow Israeli pianist Ruslan Sirota. None of these credits is individually verified against a dated release or reviewed performance beyond his own biography; see robert-anchipolovsky_notes.md. Since 2002, per the same biography, he has also made regular appearances on Israeli television. Curtis Fuller has described him as "a soulful musician with super technique," and Larry Willis has called him "one of the greatest alto players."

Anchipolovsky's playing is built directly on the Charlie Parker lineage, filtered through four years of one-on-one study with Phil Woods, one of the players most closely identified with carrying that lineage forward after Parker's death; his second cited early influence, John Coltrane, is what first pulled him from clarinet to saxophone as a teenager. His own promotional materials describe him as "passionately committed to traditional mainstream jazz," and his working career bears that out: rather than original composition or fusion, his signature format is the single-legend tribute program, a different classic songbook (Gershwin, Parker, Mulligan, Getz, Ben Webster and Oscar Peterson, the Great American Songbook) built fresh for each date. He doubles across the saxophone family and clarinet, switching instrument to match the program, most visibly picking up baritone saxophone specifically for his Gerry Mulligan tribute dates, since Mulligan was himself primarily a baritone player.