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In addition to Obergefell v. Hodges, in which the court will decide whether the Constitution guarantees a right to same-sex marriage, the justices will rule this term on cases regarding the health care law, religious freedom in prisons and the use of lethal injections. Here’s a useful guide to the term’s major decisions.

Does the 14th Amendment require a state to recognize a marriage between two people of the same sex when their marriage was lawfully licensed and performed out-of-state? Listen to the full audio and come back later for an analysis of key exchanges.

Arguing that states should not be required to recognize marriages performed outside their borders is Joseph F. Whalen, an associate solicitor general for the state of Tennessee.

Like Mary L. Bonauto, Mr. Whalen argued before the Supreme Court for the first time on Tuesday. Mr. Whalen started his career in Massachusetts before moving to Tennessee. He attended Boston University School of Law.

An hour before arguments on the same-sex marriage case were due to be finished, Democrats were busy trying to find the political advantage in their position.

Representative Debbie Wasserman Schultz, chairman of the Democratic National Committee, noted that 70 percent of Americans live in a state where same-sex marriage is legal and that the United States had come too far on the issue to turn back. She pointed to Republicans who were considering presidential bids as forces against progress.

“As Democrats stand with families and fight for fairness, Republicans like Jeb Bush, Marco Rubio, Scott Walker and the rest of the presidential hopefuls want to take us backwards,” Ms. Wasserman Schultz said.

Calling on the Supreme Court to end discrimination under the law, Ms. Wasserman Schultz argued that “love is love and it’s the right thing to do.”

The second argument has concluded. Adam Liptak, the Times Supreme Court correspondent is out of the court room and will be provide updates in the stream below and in this article.

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The second part of the arguments looked at whether the 14th Amendment protects same-sex couples when a marriage was lawfully performed in a different state.

Douglas Hallward-Driemeier had 30 minutes to make that case.

Besides Donald B. Verrilli Jr., Mr. Hallward-Driemeier has the most Supreme Court experience of the lawyers arguing on Tuesday. He has argued at the Supreme Court 15 times, both as a United States assistant to the solicitor general from 2004 to 2009 and in the private sector.

Mr. Hallward-Driemeier attended DePauw University as an undergraduate, Oxford University as a Rhodes scholar and Harvard Law School.

Victor Holmes and Mark Phariss, plaintiffs in a same-sex marriage case out of Texas, have already booked the venue for their November wedding. All they need is for the Supreme Court to deem marriages like theirs a constitutional right.

Mr. Holmes, 45, a retired major who served nearly 23 years in the Air Force, wore his uniform on Tuesday as they stood outside the court, waiting to witness the oral arguments that could confirm or upend their wedding plans.

“I can’t tell you how large my heart feels for this to happen,” he said.

The court holds a minimum of 50 seats for members of the public, and with such a big case, that was likely to be it. As the 10 a.m. start approached, he and Mr. Phariss, 55, held the precarious 49th and 50th spots in line, having had a combination of volunteers and paid-linestanders hold their places.

That is, until an alleged line-jumper snuck in, bumping Mr. Phariss, who is a lawyer. With no recourse — the waiting system depends on an honor code — the couple decided that Mr. Holmes would go in alone.

“The important thing is the result,” Mr. Phariss said, tears welling in his eyes.

Does the 14th Amendment require a state to license a marriage between two people of the same sex? Listen to the full audio and come back later for an analysis of key exchanges.

Part of the oral arguments focused on the question of whether children are better off if they are raised by a heterosexual couple. Donald B. Verrilli Jr., the solicitor general for the United States, argued there are already “hundreds of thousands of children” living with same-sex couples and that “all of the evidence” suggests that those children have not been harmed. Justice Scalia interrupted, saying that “some of the briefs contradicted that.”

During the earlier argument, Justice Samuel A. Alito Jr. raised the possibility that allowing same-sex marriages could open the door for arguments by advocates of polygamy to say the Constitution guarantees them the right to marry.

Justice Antonin Scalia offered a hypothetical situation involving four people who tried to marry, two men and two women. “They are all consenting adults. Highly educated. They are all lawyers,” he said, prompting laughter in the courtroom. “What would be the logic of denying them the same right?” Justice Scalia asked.

The potential Republican presidential candidates are struggling to adjust to a rapidly changing legal, political and cultural landscape this primary season, as the Supreme Court hears arguments on whether same-sex marriage is a constitutionally protected right.

Once a winning primary issue as well as a powerful wedge issue wielded against Democrats, opposing same-sex marriage has grown far more complicated for Republicans. While it could offer conservative candidates a way to break through a crowded primary field, it looms as a liability with general election voters, particularly independent ones, who are more supportive of same-sex marriage than more conservative Republicans.

The field of declared and presumed Republican candidates is united in opposing same-sex marriages. The only difference among them is the extent they will choose to emphasize the issue through the coming year.

Here’s a look at where the likely Republican candidates stand on gay marriage.

An ABC News/Washington Post poll conducted this month found Americans’ support for same-sex marriage at 61 percent, the highest level of support recorded by ABC/Post polls to date.

Similarly, 61 percent say they oppose allowing states to prohibit same-sex marriages, and 62 percent support requiring states to recognize same-sex marriages performed in other states.

There continue to be large differences in views of the issue by age. The poll found support for same-sex marriage at 78 percent among adults under age 30, compared with 46 percent among those 65 and older. And partisanship is a factor: While 76 percent of Democrats and 66 percent of independents support same-sex marriage, support drops to 34 percent among Republicans.