Democratic candidate Hillary Clinton on Monday night told her supporters after the Iowa caucuses that she's "breathing a big sigh of relief," but she shouldn't rest too easily.

Just a few months ago, when Clinton was dominating the Democratic field, few would have expected Sen. Bernie Sanders, the Democratic socialist from Vermont, to virtually tie the former secretary of state in Iowa. Yet heading into the February 9 New Hampshire primaries, Sanders is still a formidable contender.

The race continues on the Republican side as well: Monday night's outcome in Iowa ratchets up the race between Sens. Ted Cruz and Marco Rubio to emerge as the strongest alternative to national GOP front-runner Donald Trump.

On both sides, the number of competitors was winnowed down Monday night: After clocking in at just 1 percent in Iowa among Democrats, former Maryland Gov. Martin O'Malley dropped out. On the Republican side, former Arkansas Gov. Mike Huckabee dropped out of the running. Yet the overall dynamics of both the Republican and Democratic primaries remain relatively unchanged.

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The Clinton campaign issued a statement acknowledging the razor-thin margin between her and Sanders but saying it was still big enough that she had won. Sanders' campaign called the results a "virtual draw." Iowa's Democratic Party declared the race "historically close," but didn't declare a victor.

Had Clinton won a more decisive victory, she could have seriously dampened voter enthusiasm for Sanders. Instead, the two Democratic candidates are heading to a state where Sanders should have an advantage as a senator from Vermont.

"I know that usually New Hampshire votes for a neighbor, and [Sanders is] their neighbor, so I get that," Clinton said on CBS This Morning on Monday. "But I think I have the better plans, I think I have the better understanding of what it will take to beat the Republicans."

Sanders on Monday night reminded voters that his campaign is driven by grassroots support: Referencing his strong fundraising numbers he asked, "You know what the average contribution was? It was $27!"

While Sanders will be competitive against Clinton in New Hampshire, he faces the same hurdle that's been looming over his campaign throughout the race: He must build up support for his populist movement in the more moderate, demographically diverse states that vote later in the primary process.

Similarly, on the Republican side, it remains unclear whether Trump's populist support will carry him to the general election. His second-place finish in Iowa undercuts the businessman's claims that he always wins. "We're going to win so much, you're going to get so tired of winning," Trump told his supporters in Cedar Rapids on Monday ahead of the caucuses. Another loss in New Hampshire could significantly slow down his campaign.

Ted Cruz, meanwhile, finished first in Iowa, holding onto his status as Trump's biggest challenger. The Texan senator's data-driven, 99-county strategy in Iowa paid off, showing that his well-organized campaign could overcome Trump's celebrity status.

Marco Rubio came in a respectable third place, just a point behind Trump, winning more support (at 23 percent) than some may have expected him to. As Rubio looks more like a viable Trump alternative, the other Republicans aiming for a strong finish in New Hampshire -- including Jeb Bush, Chris Christie and John Kasich -- are likely to turn the heat up against the Florida senator. Bush and Christie in particular must do well in New Hampshire, after their hours logged in Iowa failed to translate into caucus support.

The current match-up between Rubio and Cruz mirrors the dynamic between the two senators in recent Republican debates -- including the Des Moines debate last week that Trump didn't bother to attend.