Very occasionally, the U.S. Congress will authorize an "amnesty" for undocumented immigrants (also called illegal aliens)—that is, a pardon for unlawful status, and a path to lawful permanent residence (a green card).
Such an amnesty program was offered in the late 1980s, for example. The requirements included that applicants prove they had been living or working in the U.S. for a certain length of time and had good moral character. They first received temporary status, then, after 18 months, could become eligible for green cards, provided they demonstrated that they could speak English.
In recent years, U.S. lawmakers have proposed various bills offering amnesty-like paths to a green card. As of early 2021, Congress is considering immigration reform, including a path to citizenship for undocumented persons, but this is far from a done deal.
In the meantime, some existing laws might allow something similar in rare cases, but not a true, mass amnesty.
A remedy called Non-LPR Cancellation of Removal allows non-citizens who have already been placed in removal (deportation) proceedings to ask the judge to grant a green card, provided that they have been living ("continuously physically present") in the U.S. for at least ten years; their deportation from the U.S. would cause "exceptional and extremely unusual hardship" to qualifying relatives, who are U.S. citizens or lawful permanent residents (LPRs), they have "good moral character," and they have not been convicted of certain crimes or violated certain laws.
This cannot be applied for affirmatively, however. One must be in immigration court proceedings first, presumably either after an arrest by U.S. immigration authorities or denial of some other form of immigration application.
There is a program temporarily in place providing for the deferred deportation of certain people who came to the U.S. as children and meet several guidelines (known as Deferred Action for Childhood Arrivals or DACA). This is not a law, but an Executive Order implemented by President Barack Obama. Eligible applicants receive a U.S. work permit.
In September of 2017, the Trump Administration announced its intention to terminate this program; then this effort was partially blocked by the courts; then the Biden Harris administration attempted to bring DACA back (along with proposing legislation that would offer a long-term solution); then a Texas court blocked all new DACA applications. Thus, as of summer 2021, submitting a new DACA application is impossible (although renewals remain an option for those already holding DACA).
For procedural help if and when DACA becomes available again, see Deferred Action for Childhood Arrivals (DACA): Application Process.
The chances of an upcoming amnesty or path to a green card depend on the U.S. Congress. Keep your eyes on the news and the immigration updates section of Nolo's website for changes.
Beware of the many scammers or fake lawyers who urge immigrants to pay to submit an application during a time when no such application exists. Whenever a new bill comes up in Congress, and its opponents claim (often inappropriately) that it is an amnesty, the scammers tend to seize the opportunity to start collecting money and filling out fake "applications" on immigrants behalf.