To reach the 270 electoral votes it takes to win the White House, the businessman and reality TV star will have to carry a number of states that have not voted Republican in well over a generation, while prevailing in several battlegrounds where, polls show, he starts behind.
He must also defend states the GOP has reliably counted on for decades.
"It's a very steep slope to climb," said William Frey, a demographer at the Brookings Institute in Washington, who has closely studied the political composition of the 50 states.
Ordinarily, Republicans might have greater cause for optimism.
President Obama's approval has risen throughout the year and now sits above 50% in many polls, which helps Democrats. But economic growth, while steady, has been unspectacular, which doesn't give the party or its nominee much lift. A disappointing jobs report Friday underscored that trend.
Democrats face another significant disadvantage: Their nominee, most likely Hillary Clinton, will be seeking the party's third consecutive term in the White House, something voters rarely grant.
Put those factors together and forecasts might normally suggest a slight Republican edge in November, said Alan Abramowitz, an Emory University political scientist who has written extensively on presidential elections.
"But," he added, "that's not factoring in Donald Trump."
If any election shows the danger of relying on precedent, it is this one. Few took Trump seriously when he announced he would compete for the Republican nomination against several of the party's most highly regarded prospects, including Gov. Scott Walker of Wisconsin, Sen. Marco Rubio of Florida and the state's former governor Jeb Bush.
Sizable shifts in the electoral map, however, are rare. All but 10 states have voted for the same party in every presidential race since 2000. Based on recent performance, Democrats start out the fall contest with a considerable advantage.
The Democrats have won 18 states and the District of Columbia in each of the last six presidential elections. Anchored by California and Trump's native New York, those states offer 242 electoral votes. Democrats lead in those states plus others, bringing their total to 253.
There are 13 states that have gone Republican in every presidential race since 1992, adding up to 102 electoral votes. Several other states lean strongly toward the GOP, raising the total to 191.
To reach the White House, Trump will have to greatly expand the competitive map. Demographic shifts and the Manhattan mogul's dismal standing with broad swaths of the electorate will make that difficult.
Trump says he can generate a significantly higher turnout of white blue-collar voters, boosting him in Democratic-leaning industrial states, but "two can play that game," said Whit Ayres, a Republican pollster who worked for Rubio's campaign.
"Given his incredibly derisive comments about Mexicans and other immigrants, it's reasonable to think that there would be an enormous turnout of Hispanics to stop Donald Trump," Ayres said.
If so, Nevada and Colorado — which were expected to be major battlegrounds — could move beyond Trump's reach. So would Florida, a perennial toss-up.
The nation's most populous swing state has moved slowly but steadily toward Democrats since the 2000 election ended in a chaotic tie. The 50-50 electorate that year reflected "a Florida that's gone now," said David Johnson, the former executive director of the state GOP.
The percentage of whites in the state has declined steadily and huge immigration from economically strapped Puerto Rico has changed Florida's Latino population, once primarily Cuban, to a more Democratic-leaning electorate.
In 2012, Johnson noted, Republican Mitt Romney carried about 40% of the state's Latino voters and still lost to President Obama by a percentage point. This time around, Johnson said, Trump would do well to win a quarter of Florida's Latinos.
"That's not a valid mix to win," he said. "You have to have a better message than 'The Hispanics love me,' because the numbers show that they don't."
The most likely route to a Trump victory runs through the nation's historical industrial belt, from Pennsylvania on to Ohio, Michigan and Wisconsin. He would probably need to win each of those states to prevail; all start in the Democratic column, save Ohio, which is another perennial battleground.
By emphasizing trade issues in a region that has suffered a steep decline in manufacturing jobs, Trump hopes to spur an enormous turnout of white working-class voters, attracting many who might have sat out previous elections, or voted Democratic.
A good test will be Pennsylvania, where Trump won last month's primary in a 35-point landslide over Texas Sen. Ted Cruz.
"Can Trump win? If the election were held today, the answer is clearly no," said G. Terry Madonna, a pollster and professor of public affairs at Franklin and Marshall College in Lancaster, Pa. Currently, whatever gains Trump makes among white voters in former steel-mill towns are more than offset by losses among moderates in the Philadelphia suburbs, Madonna said.
"But I don't think anyone can say definitively what will happen in the fall," he added.
Apart from his struggle in swing states and Democratic strongholds, Trump may have to fight just to hang on in states that Republicans usually take for granted.
Arizona and Georgia both have large and growing minority populations that could make them competitive in November. Missouri, with one foot in the South and the other in the Midwest, is another state that Obama lost twice that Clinton might be able to put into play.
For Trump to win the White House, many in the GOP are counting on a different candidate than the bombastic, insult-hurling personality who bulled his way to the party nomination.
Dick Wadhams, a Republican strategist and former party chairman in Colorado, said the state can be competitive if Trump changes his tone and offers more substance.
"He's already said those things that are on the public record," Wadhams said, referring to comments that have offended women and minorities, among others. "I think he can mitigate that damage by having a forward-looking, positive agenda."
Under the best of scenarios — barring a drastic change in fortunes — Trump may not win much more than the bare minimum of states needed to claim the White House. Even that will be a challenge.
"Everything," said Frey, of the Brookings Institute, "has to work just right."
Barabak reported from San Francisco and Lauter from Washington.