WASHINGTON — Tuesday morning, the Supreme Court heard oral arguments on Hollingsworth v. Perry, otherwise known as the Proposition 8 case. Wednesday it will hear arguments in a case dealing with the Defense of Marriage Act (DOMA), which prevents federal recognition of state marriages between people of the same sex.

In both cases, the Obama administration and a whole bevy of civil-rights groups, including many Republicans, want the court to rule restrictions on same-sex marriage unconstitutional.

Do they have a shot? And what precise legal issues are at stake? Here’s what you need to know.

Who are Hollingsworth and Perry?

The official title of the case is Dennis Hollingsworth, et al., Petitioners v. Kristin M. Perry, et al. Perry, who is known to education wonks as the leader of the First Five Years Fund, is one of four plaintiffs in the case, along with her partner Sandra Stier and another couple, Paul Katami and Jeffrey Zarrillo. The two couples were denied marriage licenses in the state of California following the passage of Proposition 8, which amended the state constitution to overturn the California Supreme Court’s decision legalizing same-sex marriage.

The case was originally named Perry v. Schwarzenegger, when the latter was governor of California, and renamed Perry v. Brown when Jerry Brown became governor in 2011. However, both Schwarzenegger and Brown (as governor and previously as attorney general) declined to defend Proposition 8 s constitutionality. As a result, ProtectMarriage.com, the official sponsor of the ballot proposition, stepped in to defend the proposition. The case was subsequently renamed after Dennis Hollingsworth, a former Republican state senate minority leader and a leader of ProtectMarriage.com.

What’s happened at the initial trial?

The plaintiffs, represented by former Solicitor General Ted Olson and Microsoft prosecutor David Boies (who, coincidentally, took opposing sides in Bush v. Gore), first argued their case in front of the U.S. District Court for the Northern District of California. Vaughn Walker, the chief judge of the court, presided.

The plaintiffs called a bevy of witnesses, including historians Nancy Cott and George Chauncey, who are experts on marriage and the gay and lesbian community, respectively. Cott emphasized that marriage has never had a uniform definition, and Chauncey detailed the forms of discrimination gays and lesbians have faced in the United States historically. Stanford political scientist Gary Segura testified that no minority group has been targeted by more ballot initiatives than the LGBT community.

Psychologists Gregory Herek and Ilan Meyer explained how legal discrimination affects the mental well-being of gay and lesbian people, while Anne Peplau explained the benefits of marriage to psychological health and the paucity of evidence suggesting that same-sex unions harm opposite-sex relationships, and Cambridge’s Michael Lamb noted there’s no good evidence that same-sex parents are any worse for children than opposite-sex parents. Then-San Diego mayor Jerry Sanders, a Republican, explained how he came to support same-sex marriage due to his lesbian daughter, and the writer Helen Zia explained her personal experiences with anti-lesbian prejudice. Finally, Edmund Egan, the chief economist for the city of San Francisco, testified that the city would benefit economically due to lower mental health costs for LGBT residents, among other factors.

The defendants offered only two witnesses. One, David Blankenhorn, was a left-leaning anti-same-sex marriage activist, who did his own side some damage by conceding under cross-examination that the American people would be “more American on the day we permitted same-sex marriage than we were on the day before.” Walker would state in his opinion that Blankenhorn did not qualify as an expert witness and that his testimony was “unreliable and entitled to essentially no weight.” After the case was decided, Blankenhorn changed his mind and now supports same-sex marriage. The other witness, Kenneth Miller, is a political scientist at Claremont McKenna College. Miller testified that the LGBT community has grown in political power in recent years.

On Aug. 4, 2010, Walker ruled for the plaintiffs, concluding that Proposition 8 “unconstitutionally burdens the exercise of the fundamental right to marry and creates an irrational classification on the basis of sexual orientation.”

What’s happened in the appeals since then?

First, the case was appealed by ProtectMarriage.com to the U.S. Court of Appeals for the Ninth Circuit, a famously liberal appeals court. Worse still for Proposition 8’s defenders, the three-judge panel chosen to hear the case included Stephen Reinhardt, one of the most liberal judges on the Ninth Circuit. Charles Cooper, the lawyer for ProtectMarriage.com, asked Reinhardt to recuse himself, citing the fact that Reinhardt’s wife is executive director of the Southern California ACLU, which has been involved in legal challenges to Proposition 8. He declined. The court also declined to vacate the District Court decision on the grounds that Walker is himself gay, and so had a personal interest in the case.

Reinhardt authored a 2-1 ruling upholding Walker’s decision. Reinhardt concluded, “Proposition 8 serves no purpose, and has no effect, other than to lessen the status and human dignity of gays and lesbians in California, and to officially reclassify their relationships and families as inferior to those of opposite-sex couples.” However, he declined to argue that all bans on same-sex marriage were unconstitutional, instead judging that Proposition 8 s revocation of a previously guaranteed right to marriage was particularly egregious.

ProtectMarriage.com appealed to the whole Ninth Circuit court, asking it to hear the case as en banc (that is, with all judges, rather than a panel of three). That request was denied. That left the Supreme Court as the final point of appeal. In December, the court granted certiorari, agreeing to hear the case.

What issues does the court have to decide on?

The substantive issues revolve around whether Proposition 8, or bans on same-sex marriage generally, violate the 14th Amendment’s guarantee of equal protection under the law. Usually, the courts apply one of three tests when determining whether an unconstitutional rights violation has occurred: rational-basis review, intermediate scrutiny or strict scrutiny.

Strict scrutiny, usually reserved for cases involving race, requires the state to prove that the government has a compelling interest in the subject, that the law in question is narrowly tailored to suit that interest, and that the law in question is the least restrictive possible way to serve that interest. Intermediate scrutiny, which tends to be used in cases involving sex or gender-based discrimination, merely requires the government to show that the law in question serves a government interest through means that are substantively related to that interest. Rational-basis review, which has historically been the standard applied to discrimination on the basis of sexual orientation, simply requires the government to demonstrate that it has a legitimate interest in a subject and that the means chosen to further that interest is rationally related that interest.

So the first substantive decision the court has to make is which of these standards to use. Walker ruled that Proposition 8 failed even rational basis review, as did Reinhardt. In Romer v. Evans, the most important equal protection precedent for gay-rights cases, the Supreme Court, led by Anthony Kennedy, applied rational basis review. So it’s likely that if the court overturns Proposition 8, it will rule that either all same-sex marriage bans, or else just the reversal of previously guaranteed rights to marriage, fail rational-basis review. Or, if the court fails to overturn the law, it will likely rule that there is a rational basis for banning same-sex marriage or reversing previous rulings allowing it.

Less likely is that the court will rule there’s a rational basis for Proposition 8 but that it fails intermediate scrutiny and is thus unconstitutional; that would be a historic application of a stricter standard to cases involving sexual orientation. It could also rule that there’s rational basis for banning same-sex marriage (or remain agnostic on that question) but that there isn’t such basis for offering civil unions but not marriage. That would overturn the marriage ban in California and similar bans in states like New Jersey that offer domestic partnerships, but leave other marriage bans untouched.

However, it’s possible for the court to dodge the substantive issues completely. Olson and Boies argue that ProtectMarriage.com and Hollingsworth lack legal standing to defend Proposition 8, as they would not be harmed if same-sex marriage were allowed in California. If the court agrees, it could affirm Reinhardt’s ruling against Proposition 8, effectively legalizing same-sex marriage in California, without spreading the ruling beyond California. Reinhardt’s narrow ruling against Proposition 8 offers another avenue to that same end result.

That answer was super-long. What are the actual possible outcomes?

The court could (a) rule all same-sex marriage bans unconstitutional or (b) rule Proposition 8 unconstitutional but remain agnostic on the constitutionality of same-sex marriage bans generally or (c) rule that you can’t offer civil unions without offering marriage rights too, while leaving marriage bans elsewhere untouched but affecting states like New Jersey that offers unions but not marriage, or (d) rule that Hollingsworth and ProtectMarriage.com lack standing, uphold the lower court ruling overturning Proposition 8, and take no stand on constitutionality or (e) rule that Hollingsworth and ProtectMarriage.com have standing, and that Proposition 8 and/or same-sex marriage bans generally are constitutional, reversing the lower court and setting a precedent that states can constitutionally ban same-sex unions.

All right, I think I got it. So what’s the DOMA case about?

United States v. Windsor concerns Edith Windsor, who was widowed when her wife Thea Spyer died in 2009. Windsor and Spyer were married in 2007 in Canada after being partners for 40 years. Windsor was forced to pay $363,053 in estate tax on Spyer’s estate, which she argues she would not have to pay if she had been Spyer’s husband. Thus, she claims, the Defense of Marriage Act, which prevents her from being considered Spyer’s spouse for the purposes of federal taxes, literally cost her $363,053.

The Obama administration has declined to defend DOMA, and so the Bipartisan Legal Advisory Group (BLAG), a standing organization in Congress, took over the law’s defense at the instruction of House Speaker John Boehner, R-Ohio. The U.S. District Court for the Southern District of New York ruled in June that DOMA’s definition of marriage as between a man and a woman lacked a rational basis, and ordered damages of $363,053 paid to Windsor. In October, the Second Circuit Court of Appeals concurred, with a panel ruling 2-1 for Windsor. Now the case goes to the Supreme Court.

What issues does the Court have to decide on there?

Three. The first is the equal protection issue, which is much the same in content as in the Proposition 8 case. The second is whether the fact that the executive branch agrees with Windsor means that there isn’t a real controversy in this case, meaning the court doesn’t have jurisdiction. The third is whether BLAG would be harmed by DOMA being overturned, and thus whether it has standing to defend the law (a friend-of-the-court brief by Harvard professor Vicki Jackson argues that even Congress doesn’t have standing, and even if it did, BLAG wouldn’t).

A ruling that focused on the second two issues would affirm the lower courts’ rulings and effectively void DOMA only in New York, or other states where courts have ruled it unconstitutional. But a substantive ruling would void DOMA everywhere.

Listen to Tuesday’s oral arguments on Hollingsworth v. Perry