The powerful political groups known as super PACs, whose heavy spending has become a significant factor in the presidential race, are also beginning to play a role in congressional races around the country. The groups have set off a scramble among candidates in both parties, who are now struggling to cope with a flood of negative ads run by organizations that are outside their direct control.

Targets of super PAC money in recent months include at least two dozen pivotal House districts around the country, along with high-profile Senate races in states such as Massachusetts, Ohio, Utah and Indiana, according to Federal Election Commission data and interviews with political strategists.

In Oregon’s 1st District, which is holding a mail-only special election on Tuesday to replace disgraced Democrat David Wu, Republican candidate Rob Cornilles has been bombarded with $1.8 million in ads and mailings from the Democratic Party and allied outside groups.

House Majority, Democratic super PAC, attacks Cornilles for allegations related to his sports consulting business: “His company didn’t pay federal taxes for nine months,” says one television ad. “He had to pay back wages to trainees who say they worked 11 weeks without pay.”

Cornilles condemns the attacks as “lies and distortions,” but said he still supports the court rulings that led to the rise of super PACs — and made it easier to pour more money into elections such as his.

“I don’t fault them for using the speech they want to exercise,” said Cornilles, who is running against Democrat Suzanne Bonamici. “We’re going to win this thing despite the negativity and I think, in some respects, because of it. People are disgusted by it.”

In the Massachusetts race between Sen. Scott Brown (R) and Democrat Elizabeth Warren, attacks by outside groups have been so relentless that the two candidates signed an agreement last week aimed at halting the onslaught.

Under the deal, the candidates agreed to pay a penalty that would go to charity if outside groups supporting them buy either positive or negative advertising. The two campaigns acknowledge, however, that their goal of eliminating the ads is difficult to enforce, since by definition independent groups cannot coordinate activities with candidates.

Money from non-party groups has long played a major role in U.S. politics, but a series of recent decisions, including the 2010 Supreme Court ruling in Citizens United v. FEC, have made it dramatically easier for corporations, unions and wealthy individuals to have a direct financial influence on elections. In the GOP presidential race, for example, two $5 million checks from a casino magnate and his wife to a super PAC have played a crucial role in allowing Newt Gingrich to be competitive.

The similar dynamic in congressional races is unfolding as super PACs, nonprofit groups and others join with party committees to drive spending.

One conservative super PAC, Club for Growth Action, has been running ads against Republican primary candidates it considers too moderate, including Rep. Fred Upton (R-Mich.) and Senate hopeful Tommy Thompson in Wisconsin. In Texas, the group is running $500,000 worth of television ads against Lt. Gov. David Dewhurst (R), who is vying against Club for Growth pick Ted Cruz for a Senate seat.

Club for Growth also has a regular PAC to dole out campaign money and a nonprofit that runs political issue ads.

“Our super PAC is an extremely useful tool for educating voters about the records of liberal or moderate Republicans in Republican primaries,” said Club for Growth spokesman Barney Keller. “We intend to aggressively use it this cycle to make sure that the most pro-growth candidate wins. It’s a much more powerful tool for spreading our message.”

Following court and FEC decisions , super PACs played a crucial role in the 2010 midterms, often catching vulnerable incumbents unaware. This election cycle, some lawmakers already have one or more super PACs devoted just to them, not to mention groups dedicated to electing Democrats or Republicans to one of the two chambers.

In Southern California, Rep. Howard Berman (D) faces a particularly unusual intraparty primary race in June with Rep. Brad Sherman (D) because of redrawn district maps. Two separate super PACs have signaled their support for Berman, though neither has reported any spending yet.

Perhaps not surprisingly, some of the largest expenditures by outside groups are in the most contentious races for the Senate, which has a good chance of flipping to Republican control in November. The expensive Massachusetts race, for example, has already seen millions of dollars in advertising by super PACs and others, most of it favoring Warren.

In Ohio, Democrats calculate that Sen. Sherrod Brown (D) has been hit with more than $3 million in attack ads by the U.S. Chamber of Commerce, Crossroads GPS and other Republican-leaning groups in recent months. Utah Sen. Orrin Hatch (R) is fending off primary attacks from a super PAC run by FreedomWorks, a Washington tea party group, which reported spending $120,000 on ads against him this month, FEC records show.

Wu’s former district in Oregon, which stretches from the west side of Portland to the Pacific Coast, leans Democratic and is predicted by The Cook Political Report to vote in favor of Bonamici on Tuesday. But given Wu’s resignation amid sexual harassment allegations, Democrats and their allies have spent heavily to try to keep the seat in their column.

The Democratic Congressional Campaign Committee has spent about $1.3 million, along with $300,000 by House Majority PAC and nearly $200,000 by Women Vote!, a super PAC created by the Emily’s List women’s group, FEC records show.

Cornilles, the Republican candidate, has criticized Bonamici for opposing Citizens United while benefiting from outside spending in the race. Bonamici did not return a call seeking comment.

Ali Lapp, House Majority PAC’s executive director, said the Oregon race provides a blueprint of sorts for other races later in the year, showing that independent liberal groups can work together to benefit a Democratic candidate. She said her super PAC decided to get involved in the race to err on the side of caution.

“Given the volatility of the special elections we’ve seen, we just wanted to make sure that we did what we could in a supporting role,” Lapp said.