When the offer came to play Tevye in “Fiddler on the Roof,” Danny Burstein was flattered, sure. Also surprised. Because when you ask the veteran actor if he ever expected to play the part, one word quickly comes to mind.

“Never!”

And that’s not modesty talking.

“I always thought Tevye was a role for someone older — also a star,” he says. “But, I’m quite honored to be here, let me tell you.”

Burstein, 51, was a little off with the age thing. There have been younger Tevyes — the legendary Zero Mostel was 49 at the start of the first Broadway production (which ran from 1964 to 1972) and Topol (star of the 1971 film version) was 36.

But, yes, on Broadway, Tevyes tend to be played by stars. Casting Burstein, who’s well-liked in the theater community but not yet a household name, is just one of the things that makes this “Fiddler” — opening at the Broadway Theatre on Dec. 20 — unique.

For those keeping score, this latest “Fiddler” — directed by Bartlett Sher and co-starring Jessica Hecht as Tevye’s wife, Golde — marks the show’s sixth Broadway production.

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For Burstein, a five-time Tony nominee recently seen in “Cabaret” and “Follies,” it’s his 16th Broadway gig — and third “Fiddler.” The Flushing native initially encountered the show at age 16, cast in the chorus of a production by the Diocesan Theater Guild, a community theater group in Queens and Brooklyn. He later performed in a regional production in St. Louis.

“Instead of a piano and drums, we had a full orchestra,” he notes, “I fell in love with the show.”

Yeah, get in line.

The show’s popularity is uncanny, having been performed all over the world in the last half century, from Hofstra University (earlier this fall) and Westbury Music Fair (in the 1970s) to Germany and Japan. Not bad for a show that had trouble finding initial financing because producers feared it was, well, “too Jewish.”

The musical, with music by Jerry Bock, lyrics by Sheldon Harnick and book by Joseph Stein, is based on Sholem Aleichem’s short stories of Tevye, a humble dairyman in a shtetl, or small Jewish village, in early 1900s Russia, struggling to support his wife and five daughters amid poverty and prejudice. The tales have long inspired storytellers: A Yiddish-language film, “Tevye,” was shot on a Long Island potato farm east of Jericho in 1939, and in 1953 an English-language adaptation debuted Off-Broadway. (“Don’t think because it’s about Jews you won’t like it,” urged Eleanor Roosevelt in her newspaper column.)

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The musical’s creators took pains to retain those Jewish roots. Bock employed rhythms and melodies akin to cantorial music and Hasidic chants. And Harnick and Stein scoured books like “Life Is with People,” a 1952 text about Eastern European shtetls. “I learned a lot from that book about life in the Old Country,” Harnick recalls.

A chapter on arranged marriages, for instance, notes how young women often gossiped about potential pairs (cue the “Matchmaker” song) and that, in shtetls, “first you marry, then you love,” an idea not far from the “Do You Love Me?” duet sung by Tevye and Golde.

The show’s original star, Mostel, also weighed in.

Harnick recalls how he’d considered cutting the last verse of “If I Were a Rich Man,” when the tempo slows and Tevye stops joking to sing about spiritual concerns, yearning for more time to pray.

“Zero screamed at me, ‘Don’t you dare cut that! The rest of it is amusing but that’s the man at the end. Please, leave that in.’ And he was right.”

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Jerome Robbins, the show’s original director and choreographer, immersed himself in Hasidic culture, even crashing weddings in Borough Park, Brooklyn, mining for authentic dance moves. Subsequent professional revivals have had to use Robbins’ staging — until now. The current revival has been granted some leeway, bringing in Israeli choreographer Hofesh Shechter to rework the dances, giving them a more athletic (and thus more current) look.

This may be sacrilege to Robbins fans, but, rest assured, the famed “Bottle Dance” remains intact — it’s just what Shechter calls “Bottle Dance 2.0.”

Audiences’ devotion to such elements underscores the remarkable way in which this show transcends cultures. “Fiddler” isn’t just about Jews in Eastern Europe. It’s about family, fulfillment and the generation gap. “That’s why ‘Fiddler’ will never go out of style,” says Burstein. “It touches on themes we can all relate to and does so in a beautiful, entertaining way.”

For him, Tevye feels modern — progressive, even — bending old traditions for the sake of his children, and arguing with God. “He cajoles, he pleads, he yells at God — he dares to question,” Burstein explains.

Add to that the current Syrian refugee crisis, an immigration problem the likes of which we haven’t seen since post-World War II, and the show feels more relevant than ever before.

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“The last scene,” says Harnick, “the exodus, where all the people are leaving with their bundles — it looks like something right out of the daily newspaper.”