President Trump has moved to roll back the government programs that protect certain immigrants in the United States from deportation, prompting several court challenges amid his administration’s broader immigration crackdown.

Over the past several months, Mr. Trump has revoked the legal status afforded to some Ukrainians displaced by Russia’s invasion and Afghan citizens who helped the American war effort in their country. He has also canceled the protected status of Cubans, Haitians, Nicaraguans and Venezuelans who fled instability and political violence back home — potentially leaving them vulnerable to deportation.

Perhaps most prominent has been Mr. Trump’s targeting of nearly 350,000 Venezuelan immigrants who had been allowed to stay under a program known as Temporary Protected Status. In response to the administration’s emergency application, the Supreme Court on Monday allowed the government to move forward with plans to lift those protections — at least for now.

But other cases involving immigrants with protected status are moving forward, as well, with thousands of people in limbo. Here’s what to know about the major challenges to Mr. Trump’s actions, and where things stand.

Cubans, Haitians, Nicaraguans, Venezuelans: Noem v. Doe

The Supreme Court has not yet released a decision on another emergency application that could affect Venezuelan immigrants. In that case, the Trump administration has asked the court to allow it to revoke deportation protections for migrants from four troubled countries: Cuba, Haiti, Nicaragua and Venezuela.

A lower court judge issued an emergency order in April blocking the termination of legal status of many of the more than 500,000 migrants who entered the country under the program.