Suarez clearly relishes his new-found political standing, which sometimes evinces the stubborn streak that drove him to spend weeks in 1983 searching for his car that was stolen. After two months, Suarez spotted his blue Impala parked in front of an apartment, got his keys, and "re-stole" it.
There are those who whisper that Suarez's clout has gone to his head.
He angered some legislators this year with a strongly worded, threatening letter demanding their support for a 2% food and beverages tax (which later passed).
Told someone is canceling an appointment with him, Suarez says: "Tell him that's the last appointment he gets with me." Told U.S. Rep. Bill Nelson, a Democrat, has been courting influential Cubans for support of his 1990 gubernatorial bid, Suarez snorts: "He hasn't talked to me yet."
While voicing ambitions beyond Miami, Suarez says he still has much left to do here. He streamlined the government of what he called "the city that doesn't work" in his 1985 campaign that saw the defeat of incumbent Maurice Ferre. But as the January riots showed, work remains to be done in the inner cities he pledged to make a priority.
Suarez has sheaves of statistics to show progress made in adding housing, spurring businesses and improving services in the impoverished black neighborhoods, but he concedes that more needs to be done to lift the spirits and hopes there.
Suarez drew praise by walking virtually unprotected through riot areas, and his ratings among blacks were strong in the poll taken by political consultant Sergio Bendixen. He said Suarez was in a unique position among Cuban-American politicians because of his cross-ethnic appeal.
Getting away from Miami's ethnic-bloc politics has been a major goal. He opened a press conference with Latino journalists by warning there were two questions he wouldn't answer: "What's it like being the first Cuban mayor?" and "How do you compare yourself with Henry Cisneros?"
"I'd rather just talk about being mayor of Miami," Suarez said.
That job, though, turned out to include being a special figure in Latin America, just as Ferre had been and Suarez had scoffed at. Nearly all Latins view Miami as something of a suburb, he explained, a place to shop, to invest, to get medical service, to cut deals, to catch up on what's going on in the region. The Miami news media is followed throughout Latin America, Suarez said.
"I did take a low profile on international issues at first. I wanted to batten down the hatches and get the city moving," Suarez says. "Now I realize you can do it totally parallel to being mayor."
For now, he will run for reelection in November, this time to a four-year term after two two-year terms. He hopes to see voter approval of a strong-mayor system, and also a strong-county-mayor system.
He, his wife and four young children are planted firmly in Miami, Suarez says, for the near future.
"It's a very exciting city," concluded its not-so-dull leader.