On Friday, the Supreme Court decided to tackle the case of the Little Sisters of the Poor, a group of nuns who believe, along with some priests, a Roman Catholic Archdiocese, and several universities, that the government is compelling them to violate their beliefs. Their claim: The so-called birth-control mandate of the Affordable Care Act places a burden on their religious exercise, even with an accommodation from the government.

The first thing to know about these cases is that they are incredibly complicated. The Court granted cert in seven different cases related to this topic, which means they’ve agreed to hear the questions in those seven cases. All of them have come through different circuit courts in the past months, where they’ve mostly lost. But, these cases are a big deal: They are the latest in a long series of challenges to this portion of the law, the most notable of which was last summer’s Hobby Lobby case, which involved for-profit employers.

The Affordable Care Act requires all U.S. insurance plans to cover 20 varieties of FDA-approved contraceptives at no cost to patients. This affects employers at both for-profit and non-profit organizations, because they have to provide coverage for contraception in their insurance plans. Immediately following the passage of the law in 2010, a number of organizations objected, saying that some of the approved forms of contraception are the equivalent of abortifacients, or drugs that cause abortion. If they refused to provide the coverage, they would face heavy fines.

So, the government set about making exceptions. Explicitly religious organizations—churches, synagogues—are exempt from this requirement. After the Supreme Court’s decision in Hobby Lobby this summer, an accommodation was created for closely held, for-profit businesses that have a religious objection—they can fill out a form and submit it to the government, which prompts a third-party organization to provide the coverage instead.

This kind of accommodation is also available to religious non-profit organizations, which includes the seven plaintiffs in these cases: the Little Sisters of the Poor; a pro-life organization called Priests for Life; representatives of several Roman Catholic organizations; and Geneva College, Southern Nazarene University, and East Texas Baptist University. But these organizations say this is insufficient for a number of technical reasons, the most important being that they believe filling out the form still amounts to complicity in providing people with contraception.

The relevant law in these cases is the Religious Freedom Restoration Act, or RFRA, a federal statue which provides a test for sorting out whether a law places an undue burden on a person or organization’s religious beliefs. The first question that has to be answered is: Does the law place a substantial burden on an organization’s religious exercise? If the answer is yes, then the second question is: Has the government found the “least-restrictive” way to accomplish its original goal in passing that law, or is there another, less restrictive way of doing so? Lawyers for the Little Sisters had also argued that the birth-control mandate is a violation of the nuns’ First Amendment rights to free exercise, but the Court ignored that; it will only evaluate their objections under the tests of RFRA. In a call with reporters, a lawyer for the Little Sisters said he thinks this could be “a signal that the Court thinks RFRA is sufficient to resolve this issue.”

These questions may seem fairly technical and nit-picky. But the way the Court rules in Little Sisters and these accompanying cases will be legally and politically significant for a few reasons. This is culmination of the third major round of legal challenges to this portion of the ACA: the initial wave of pushback from religious organizations; the challenge from for-profit employers, which culminated in Hobby Lobby; and now, objections from a number of religious non-profits.

There are many, many similar cases that have been litigated around the country that didn’t make it to the Supreme Court in time to be included with this batch of cases, such as that of Wheaton College. The Court’s eventual decision will affect the fate of those cases, and potentially the fate of a number of religious non-profits, such as Notre Dame, that lost their legal battles in lower courts and have reluctantly conceded to providing coverage. A victory for Little Sisters and co. could potentially open the path for further legal challenges from for-profit employers with religious objections to contraception; right now, these companies get an accommodation that works very similarly to the accommodation for religious non-profits. A victory for the religious non-profits would also be another blow to the Obama administration, which has pushed back hard on these various exemptions. As the lawyer for Little Sisters put it on Friday, “They’ve been on a bit of a losing streak on this issue.”

A loss would be equally significant. These cases are sort of similar to a game of chicken: As the cases creep closer and closer to their conclusion, the stakes become starker. If the Little Sisters and co. take their objections to the highest Court in the land and are turned away, will they still refuse to provide the coverage and face fines—$75 million per year, estimated the nuns’ lawyer—or, potentially, having to shut down? Or will they, like Notre Dame and others, reluctantly give in, shelving their deeply held religious beliefs for the greater good of continuing to feed the poor, care for the elderly, and educate students?

Stay tuned as the Sisters take on the Supremes.