Today Hadid is still turning convention on its head–but she’s producing practical architecture at the same time. “I’m challenging typology,” she says. “I’m rethinking organization and how to reinvent civic domain.” Of course, a lot of architects could say that–whatever it means–but as the millennium turned, Hadid actually began getting her projects built. Perhaps it was because of that small, totally cool fire station she constructed for the Vitra company in Germany in the mid-’90s. Or maybe it was just that the world was finally ready for her. This month Hadid’s first building in the United States will open–not in some cultural capital like New York or L.A., but in Cincinnati. (Leave it to the heartland to be hip enough to embrace Hadid first; her next American project is an art-and-design museum in Bartlesville, Okla.) Her Contemporary Arts Center is no forbidding outpost of the esoteric. Its approach–on a prime corner of downtown–is surprisingly open and democratic. The sidewalk literally continues right into the glassed-in first story, with its concrete floor–the “urban carpet,” she calls it–inviting passersby to come in and hang out. Then, around the corner, the pavement swoops up into a curve that ingeniously becomes the building’s back wall.

The center has a history of risk taking: in 1990 its then director was famously arrest-ed (and later acquitted) for showing Robert Mapplethorpe’s photographs. Standing on the street, you think at first Hadid’s building looks quite polite–it’s in scale with neighboring structures–but at the same time it’s wonderfully dynamic. Monolithic chunks jut out and cantilever over the street corner; the heavy upper stories, of concrete and black steel panels, hover over the light-as-air glass base. “It needed to be quite subtle, but with a hint of mystery,” she says. “You don’t realize everything from the outside.” True enough: inside, the gallery spaces nestle and bump and flow into each other–unlike the blandly uniform galleries of many museums. And linking the six floors is Hadid’s most dramatic element–enormous diagonal steel beams that contain the stairs. When they crisscross, they remind you of those great 1930s photos of American factories by Charles Sheeler. Thank God Zaha doesn’t shy away from the tough stuff.

Now 52, she’s lived in London since her days in architecture school (where Koolhaas was her teacher). But she grew up in Baghdad, back when it was a cosmopolitan city and home to “every kind of minority.” The daughter of liberal Muslims–her father was an industrialist and the leader of a progressive Iraqi political party–she attended a French-speaking Roman Catholic convent school, where “the Muslim and Jewish girls could go out to play when the other girls went to chapel.” Everyone in her family went abroad to university, and, she says, “There was never a question that I would be a professional.”

In her funky London office in an old brick school building, she presides over a staff of nearly 50 people who are working on upcoming projects: a science museum in Germany, an art center in Rome, a ferry terminal for Salerno, a bridge in Abu Dhabi, a master plan for Singapore–and the museum in Bartlesville. With her mane of auburn hair, her gravelly voice and the swaths of Issey Miyake clothes she loves to wear, Zaha has been called a diva. But she’s quick to point out that no one criticizes a man for being tough, for refusing to compromise. She simply worked and waited–and now her rethinking of architecture’s most basic rules seems right on time.