Cush Jumbo. Remember the name.

Although it sounds as if it could be a circus act, this is the name of an extraordinary British actress who has been impressing around New York's theatrical edges in recent seasons. As anyone knows who saw her Marc Antony in Phyllida Lloyd's smashing all-woman "Julius Caesar" in 2013, Jumbo can do daring Shakespeare. And as big-ticket Broadway theatergoers may remember, they met her briefly this season as one of Hugh Jackman's mysterious women in "The River." But that, seriously, was not about her.

"Josephine and I," her solo play at Joe's Pub at the Public Theater, is all her. And all Josephine Baker. Jumbo wrote and stars in this smart, altogether entertaining and engaging show about a young English actress very much like her and about Baker, the breakthrough black performer who pulled herself out of a hellish St. Louis childhood to become the toast of Paris, a hero of the anti-Hitler French resistance and an international show-business legend.

This singing, dancing, acting drama is again directed by Lloyd, whose range spans from the cheerful silliness of "Mamma Mia!" to the riveting "Mary Stuart" on Broadway in 2009. In the play, which runs just under two nonstop hours, a character the program calls Girl recounts her lifelong obsession with Baker -- "the first movie star who looked like me." Girl — who, like Jumbo, has a Nigerian father and an English mother — even gave her pinky-white Tiny Tears doll an "extreme makeover" with brown paint and faux-African exotica clothes.

We know this because Jumbo has brought the doll along to agree periodically with this or that biographical or sociological detail. It would be a mistake, however, to let the doll make you expect kid's stuff. This is a grown-up story that cleverly overlaps the contemporary struggles of an actress and the rise-and-fall-and-almost-rise of Baker, who, among other adventures, was in "Shuffle Along," the first all-black Broadway show.

Jumbo, who trained as a dancer, has endless flexible arms and legs that can feel hard-wired to some English girl's idea of African dance. She can be impish and elegant, goofy and powerful, with huge eyes and a mouth that suggest a multicultural Keane painting.

In 2013, this show got her England's Evening Standard Award for "Emerging Talent." I think she has emerged.

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WHAT "Josephine and I"

WHERE Joe's Pub, Public Theater, 425 Lafayette St.

INFO $50; 212-967-7555; publictheater.org

BOTTOM LINE Breakthrough show by Cush Jumbo.