Finally, the results are in, and contributors to CNN Opinion weigh in on what they mean:

Julian Zelizer: Challenges of the second term will be immense

President Barack Obama comes away with a victory, but it won't allow him to rest easy. He pulled off the Electoral College votes that he needed, securing wins in states that Republicans once thought to be secure and preserving Democratic strength in other areas.

Mitt Romney failed to carry most of the key battleground states that were essential to his victory. The Republican failure to appeal to key constituencies, such as Latinos, has proven to be more costly with each election.

But the challenges of Obama's second term will be immense. Polls show that the electorate is unhappy. The House will remain under the control of Republicans, and Mitt Romney ran a competitive race. This is not what any incumbent hopes for, especially with a Washington that is so gridlocked. Presidents want a commanding victory, hoping for another 1936, 1964 or 1984.

Obama will need to quickly find areas of possible compromise on issues such as deficit reduction and immigration reform, where Republicans may see an incentive to negotiate. This is what Ronald Reagan did with tax reform in 1986.

Such breakthroughs have the potential to steal some of the thunder from the GOP, even if they anger members of the president's own party, allowing him to enjoy some policy victories that will strengthen his historical legacy.

Julian Zelizer is a professor of history and public affairs at Princeton University. He is the author of "Jimmy Carter" and the new book "Governing America."

Paul Begala: Obama overcomes the odds

It is hard in the moment to find the proper perspective for President Barack Obama's re-election. Political strategists like me fall back on the strategic and the tactical: this ad worked, that debate performance soared, that state's get-out-the-vote effort failed. Such analysis is inadequate to the task.

Barack Obama's 2012 re-election was infinitely more difficult than his first historic election. With a stagnant economy, a bitterly determined and zealous opposition, and the risk of sagging enthusiasm among his true believers, the odds against the president were nearly insurmountable. The economy alone would have sunk nearly any other politician. But Obama has always beaten the odds.

The president kept his party united and avoided a primary challenge, Job One for any incumbent. He assembled a remarkable team, from the high command in Chicago to the farms of Iowa and the factories of Ohio. He even reached out to his golden-tongued predecessor and a raspy-voiced rocker: Elvis and the Boss. His funny, feisty, fiery vice president was an underrated asset. And so was first lady who never stooped to the level of the critics and awes all of us who struggle to raise good kids in tough times.

The Republicans have a lot of soul-searching to do. But for now, let me congratulate our president and his brilliant team for a victory that was as hard-fought as it is well-deserved.

Paul Begala, a Democratic strategist and CNN political contributor, is senior adviser to Priorities USA Action, the biggest super PAC favoring President Barack Obama's re-election. Begala was a political consultant for Bill Clinton's presidential campaign in 1992 and was counselor to Clinton in the White House.

Anne-Marie Slaughter: Good news for rest of world

Obama's re-election was good news for the world. And judging by the congratulations and jubilation coming in on my Twitter feed from all over the globe, the world knows it. Aziz, my Moroccan hairdresser, told me Wednesday morning that his entire family (proud new members of Morocco's emerging middle class) was as jubilant as his neighbors in New Jersey.

The most important implication of Obama's victory, at least in the short term, is continuity. Given the fragility of the global economy and the number of ongoing crises and conflicts, this is no time for a sharp change in America's course.

Change is coming in China, Japan and undoubtedly a number of European countries as part of the fallout from the euro crisis; governments are new or in transition across the Middle East and North Africa; speculation continues about the health of Russia's Vladimir Putin and Turkey's Recep Erdogan. A steady hand at the helm in the White House will be more important than ever over the next four years.

Equally important, a newly elected Obama will be in a position to do a number of things that the world badly needs.

First is to get serious about the U.S. approach to climate change. The combination of his re-election, the devastation inflicted by Hurricane Sandy and the dawning awareness among even very resistant Americans that weather patterns really are matching climate scientists' predictions, and the possibility of raising much needed revenue from a carbon tax or cap-and-trade schemes all augur well for serious federal legislation at last.

Next is immigration reform, which was on Obama's 2008 agenda but now must come to a head. Republicans staring at U.S. demographic changes over the coming decades, changes that were already visible in this election, would be suicidal to block it.

Finally, the president will return to his global zero agenda, working decisively to put the world onto a clear path toward a future without nuclear weapons.

Countless additional issues, crises and conflicts await the president and his new foreign policy team. But for today, Obama's campaign slogan has actually won the day -- forward.

Anne-Marie Slaughter is a professor of politics and international affairs at Princeton University. She curates foreign policy on Twitter at: @slaughteram

Donna Brazile: A mandate to move America forward