With Joseph R. Biden Jr. edging closer to victory in the presidential race, President Trump and the Republican Party have been intensifying efforts to halt the counting of ballots and to challenge the ballots of Democratic voters in lawsuits across the country.
Nearly a dozen suits were already making their way through the courts in Nevada, Pennsylvania, Michigan and Georgia, four key states where Mr. Biden leads or has won the vote count.
But none of them appear — at least not yet — to provide Mr. Trump what he would need to prevail: the rejection of enough Democratic ballots in enough states to reverse any Biden victory.
If Mr. Trump and the Republicans cannot find those rejections through the courts, they could — and will — try to do so through recounts, but the bar is high there too.
Here are the cases currently underway or facing a potential appeal. All of them have been initiated by Mr. Trump, his party or his allies.
The Postmark Battle
Status: At the U.S. Supreme Court
In September, the Pennsylvania Supreme Court ruled that election officials could accept ballots postmarked by Election Day but arriving up to three days later. Republicans subsequently sued, and the case made it to the U.S. Supreme Court.
But in late October, the Supreme Court declined to intercede, saying it was too close to Election Day to make such a ruling, but it left open the possibility of a decision at a later date.
On Wednesday, the Trump campaign filed a motion to intervene in the case, and on Friday the Pennsylvania Republican Party sought to join the effort.
But this fight may prove fruitless, as Mr. Biden’s lead in the state is based on ballots cast by Election Day and is expected to grow. Pennsylvania’s secretary of state, Kathy Boockvar, said on Thursday that there were not large numbers of late-arriving ballots. As of Friday night, there had been no further action on this case.
Status: Pending in state court and resolved in federal court
On Thursday morning, a Pennsylvania state court handed Mr. Trump a minor victory. A judge ruled that election observers from the Trump campaign, who were allowed to stand 10 feet from the vote counting at the Philadelphia convention center, could move closer, to six feet.
By the end of the day, however, lawyers for the Trump campaign had filed an emergency petition in federal court claiming that election officials in the city were not abiding by the state court’s decision and asking that the count in Philadelphia be delayed.
At a hastily scheduled hearing Thursday night, however, Mr. Trump’s campaign admitted that “a nonzero number” of Republican observers had in fact turned up.
“Then what’s the problem?” Judge Paul S. Diamond asked.
The Trump campaign ultimately agreed to drop its request to halt the vote count after Judge Diamond, who was appointed by President George W. Bush, allowed for a total of 120 observers at the convention center — 60 for the Democrats and 60 for the Republicans. The Pennsylvania Supreme Court has not yet ruled on the question of the observers.
Status: Pending in state and federal courts
Republicans have brought suits in federal and state courts alleging that Ms. Boockvar provided improper guidance to counties by allowing them to contact voters whose mail ballots had been rejected because of errors so that those voters could fix, or “cure,” their ballots or cast provisional ballots.
Both cases focus on votes in Montgomery County, where officials say only 98 ballots might be affected.
.A judge in the federal case, who is also a George W. Bush appointee, expressed skepticism during a hearing on Wednesday about the validity of the Republicans’ challenge. A decision is pending.
Mail-in Voter ID
Status: Pending in state court
The Trump campaign has also sued Ms. Boockvar for her decision to extend by three days, to Nov. 12, the deadline by which mail-in voters must submit materials confirming their identity if they are first-time voters in certain districts. It is unclear how many votes that case would potentially affect.
Observers/Signature Matching in Clark County
Status: Pending appeal at state level
Mr. Trump’s campaign filed suit before Election Day seeking to stop the processing of mail-in ballots in Clark County, which includes Las Vegas. The campaign alleged that county officials were failing to give Republican observers adequate access to monitor mail-in ballot processing and that the county’s signature matching system violated election equal protection laws because it was not being used elsewhere in the state.
A judge denied the Trump campaign’s request earlier this week, citing a lack of evidence. An appellate court rejected Republicans’ request that it order an immediate stop to counting but agreed to hear arguments into next week.
On Thursday, Republicans indicated that they would drop their case in return for an agreement from the county to expand their observers’ access to ballot counters, but Democrats refused to agree to a dismissal, so the case is still pending. Republicans have since filed a similar suit in federal court.
Status: Resolved in federal court
In an effective extension of the state lawsuit, two Republican House candidates in Nevada filed a lawsuit on Thursday alleging that there were “lax procedures for authenticating ballots” in Clark County and that more than 3,000 ballots had been cast by inelligible voters, including some cast “on behalf of deceased voters.”
The case was assigned to Judge Andrew P. Gordon, an appointee of President Barack Obama’s, who dismissed it on Friday. The two Republican candidates who brought the case, however, can still appeal the decision.
Status: Pending in state court
It was one of the stranger claims of the election. Hours after polls closed in Arizona, a story ricocheted around online saying that dozens, maybe even hundreds, of ballots across the state had not been counted because voters had filled them out with felt-tipped Sharpies and not with ballpoint pens.
Even though cybersecurity officials from the Department of Homeland Security urged people to ignore the tale, crowds turned up outside of a polling station in Maricopa County, yelling about “SharpieGate.”
On Wednesday, Laurie Aguilera, a Maricopa County voter, filed a lawsuit with help from a conservative legal group in Indiana, the Public Interest Legal Foundation, claiming that her ballot — and those of untold others — had not been read properly by vote scanning machines because she had used a Sharpie and “the ink was bleeding through.” Ms. Aguilera has asked a judge to let all voters who filled their ballots out with Sharpies to “cure” them.
On Thursday, the Maricopa County Elections Department released a statement saying that Sharpies were in fact “recommended by the manufacturer” of the vote tabulations machines the county uses. Later that day, the state attorney general’s office issued a letter noting that the use of Sharpies in Maricopa County “did not result in disenfranchisement.”
53 Late-Arriving Ballots
Status: Tossed by county court
The Trump campaign filed a lawsuit in Georgia on Wednesday, claiming that a witness had observed 53 late-arriving ballots in Chatham County not being properly stored, potentially allowing for them to mix with timely ballots, and asked that ballot counting in the county be stopped.
But Judge James Bass, on the Chatham County Superior Court, tossed the lawsuit on Thursday, saying that there was no evidence that those 53 ballots had been received after the 7 p.m. deadline and that there was no evidence county officials had failed to comply with the law.
There was no word from the Trump campaign or its Republican allies on Friday whether an appeal was pending.
The Trump campaign filed suit on Wednesday asking a state judge to halt vote counting, alleging that its observers had been blocked from meaningful access to counting rooms. The campaign also asked for access to surveillance footage of the state’s ballot drop boxes.
A judge rejected that suit on Thursday, noting that the counting had finished. Mr. Biden won the state and maintains a lead of nearly 150,000 votes. But it is not clear whether the Trump campaign will appeal.
Georgia: Brad Raffensperger, the secretary of state, announced on Friday that the state would conduct a recount in the presidential race, saying the results would fall within the margin of a recount. “We are literally looking at a margin of less than a large high school,” Gabriel Sterling, Georgia’s voting implementation manager, said.
Wisconsin: Mr. Trump would be entitled to a recount in Wisconsin as long as the margin between him and Mr. Biden remained less than 1 percent of the vote. The state’s preliminary results show Mr. Trump trailing by about six-tenths of 1 percent.
A recount request cannot be made until all 72 of the state’s counties submit their results to the Wisconsin Elections Commission, which are due by Nov. 17. The Trump campaign would have to pay for a statewide recount unless the margin shrinks to less than one-quarter of 1 percent.
Pennsylvania: State law requires an automatic recount if the result is half a percent or less. If the margin is larger than that, Mr. Trump could still request a statewide recount, but he would have to pay for it.
Arizona: State law requires a recount if the margin is one-tenth of one percent or less — otherwise one cannot be requested.
Nevada: Any candidate or campaign can request a recount within three days of the final statewide canvass of results, regardless of the margin. There are no automatic state recounts.