It's become a perennial problem for Republicans, but not one that the party yet knows how to solve.
One recent poll showed Romney's support among African-Americans at 0 (yes, zero) percent.
In a sense, this is nothing new. As long ago as 2001, Rich Bond, a former head of the Republican National Committee, told The Washington Post: "We've taken white guys about as far as that group can go. We are in need of diversity, women, Latino, African-American, Asian."
What has changed is that minority voters now make up a large and growing share of the electorate. Between 1992 and 2008, the non-Anglo portion of the electorate doubled, to 26 percent from 13 percent, as measured by exit polls.
According to a recent National Journal analysis, Romney will need the percentage of white voters to remain at 74 percent nationwide — and he'll have to take 61 percent of that white vote — in order to win.
"This year or 2016 will be the last time Republicans can do as well as they've done in recent decades with [just] a strong showing among white voters," says Henry Olsen, vice president of the conservative American Enterprise Institute. "At some point in the not so distant future, Republicans have to start doing better among minorities or they will not win elections."
One way the party is hoping to speak to minority voters is by having minority officeholders speak to them. The GOP's convention lineup this week is loaded with high-profile minorities, including Florida Sen. Marco Rubio (Thursday), former Secretary of State Condoleezza Rice (Wednesday) and Govs. Nikki Haley of South Carolina and Brian Sandoval of Nevada (who spoke Tuesday) and Susana Martinez of New Mexico (Wednesday).
"So often, the Republican Party is characterized as being old white men," says Ellen Roberts, a Republican state senator in Colorado. "As a woman and not yet quite old, I don't agree with every part of the Republican platform, but people need to know there are others out there, whether it's someone who is African-American or Hispanic, who are often overlooked as being members of the Republican Party."
The GOP's effort to show an inclusive face in Tampa has itself triggered partisan argument. Los Angeles Mayor Antonio Villaraigosa, who will chair the Democratic National Convention next week, said at a Tampa news conference Tuesday, "You can't just trot out a brown face or a Spanish surname and expect people are going to vote for your party."
That kind of comment, along with a larger debate about whether the party's current complaints about Obama's welfare policies and ongoing comments about the president's birth certificate amount to "playing the race card," drew an angry riposte from Haley, an Indian-American.
"It's offensive to me as a woman and as a minority that Democrats can go and say, 'That party hates you,' and can get away with that," Haley told an editorial board from Gannett and USA Today on Tuesday.
Haley suggested that her party offers a welcoming home to many minority voters and is a good fit for them on issues such as the economy and jobs.
"I think we the Republicans have been mistyped as unfriendly to minorities and not receptive to minority views," says Texas Rep. Bill Flores. "We feel that the Republican platform presents the best opportunity for the success of minority families. All minority groups seek the same opportunities in the United States that the settlers from Europe did when they came over."
In a victory for the Obama administration, a three-judge federal panel on Tuesday threw out congressional and legislative redistricting plans in Texas drawn up by Republican Gov. Rick Perry and the GOP-dominated Legislature, finding that the maps discriminated against minorities.
"We are also persuaded by the totality of the evidence that the plan was enacted with discriminatory intent," wrote Judge Thomas B. Griffith, who was appointed to the District of Columbia Circuit by President George W. Bush.
Republicans have long argued that many minority voters share the party's values when it comes to issues such as same-sex marriage and abortion. Many have suggested that Democrats take minority support for granted and don't speak properly to their interests.
But it's the GOP's current platform that is largely a turnoff to minorities, suggests Alan Abramowitz, a political scientist at Emory University. "The message the party is sending out with its policies is in many ways anathema to African-Americans and Latinos," he says.
Abramowitz points to issues such as the party's hard line on illegal immigration and its promotion of voter ID laws that many minorities feel are designed to keep them from exercising their franchise.
That's a characterization Flores and other Republicans reject. "I'm troubled when people say voter ID is unfriendly to the minority community," Flores says. "What it does is preserve the sacred trust of their vote as much as anybody's vote."
The back and forth about various issues and racially polarizing rhetoric, however, has not solved the GOP's central problem: how to attract more minority support.
"In the long run, Republicans have to figure out how to talk to minority voters, or they perish," says AEI's Olsen.