Lower courts have struck down each version, but the conservative-leaning Supreme Court has given the administration hope that Trump may be able to carry out one of the most significant and divisive initiatives of his presidency.
What the president has said is a necessary step to protect the country from terrorism has been characterized by his opponents as an illegal and unconstitutional fulfillment of campaign promises to ban Muslim immigrants.
His first order, issued a week after his inauguration, created chaos at airports and prompted protests in the United States and abroad. Trump's comments and tweets about immigration and terrorism have ensured that the issue remains at the front of the national debate.
In a short order accepting the case for argument in April, the justices said they will consider the statutory justification for Trump's actions as well as charges from opponents that it is an unconstitutional restriction based on religious discrimination.
The order would also seem to further ensure a blockbuster conclusion to the court's term in June — the Supreme Court already is considering potentially landmark cases on partisan gerrymandering, privacy, unions and a clash between religious rights and gay rights.
In requesting the court provide a final answer on the travel ban, Solicitor General Noel J. Francisco said the high court must reestablish the vast authority the president wields when the nation's security is at stake.
"The courts below have overridden the President's judgments on sensitive matters of national security and foreign relations, and severely restricted the ability of this and future Presidents to protect the nation," Francisco wrote in his petition to the court.
Challengers to the ban — in this case, the state of Hawaii and others — say Trump has exceeded his legal authority.
"No prior president has attempted to implement a policy that so baldly exceeds the statutory limits on the President's power to exclude, or so nakedly violates Congress's bar on nationality-based discrimination in the issuance of immigrant visas," said Hawaii's brief asking the court to let lower-court decisions stand.
But there are indications that the Supreme Court will be more sympathetic to the administration's claim than the lower courts that have rejected it.
Last month, in an unsigned opinion, the justices said the restrictions in Trump's latest version of the ban could go into effect as envisioned while legal challenges to the merits of the decision continued.
That decision, which included noted dissents only from liberal Justices Ruth Bader Ginsburg and Sonia Sotomayor, in effect discarded a compromise the justices fashioned regarding the second version of the plan. That compromise said the ban would not affect those who could prove significant connections to the United States.
The current version of the plan imposes various restrictions on travelers from Syria, Libya, Iran, Yemen, Chad, Somalia, North Korea and Venezuela. The first six of those countries have Muslim-majority populations, and the restrictions on travelers from North Korea and Venezuela are not part of the challenge.
The court will review a unanimous ruling from a three-judge panel of the U.S. Court of Appeals for the 9th Circuit in San Francisco. That panel said the third version of the travel ban suffered from the deficiencies of the first two — that Trump had again exceeded his lawful authority and that he had not made a legally sufficient finding that entry of those blocked would be "detrimental to the interests of the United States."
Another challenge to the ban is still resting with the U.S. Court of Appeals for the 4th Circuit in Richmond. That full court is considering a ruling by a Maryland judge that the ban violates the Constitution because it is effectively a ban on Muslims. Judge Theodore D. Chuang considered Trump's statements and tweets in reaching his decision.
Despite a nudge from the Supreme Court last month to work quickly, the 4th Circuit has yet to rule. The case could be added, but the Supreme Court told the administration and Hawaii to also argue that constitutional question.
Trump's moves on the issue have been controversial from the start. After the first version was rejected by the courts, the administration replaced it with a second one. It barred entry by nationals of six overwhelmingly Muslim countries for 90 days, excluded all refugees for 120 days and capped annual refugee admissions at 50,000.
Courts put that order on hold as well. That is when the Supreme Court fashioned the compromise that permitted the ban to go into effect but granted entry to those with a significant connection to the United States, such as family members, a waiting job or an academic opportunity.
The justices were scheduled to decide the merits of that ban last fall. But the order expired before oral arguments, and the administration replaced it with the third order.
Whatever the shortcomings of the first two orders, Francisco told the court, the third followed a painstaking review process to determine which countries did not have procedures in place to screen out those who might intend to harm the United States.
The eight countries named in the current ban "do not share adequate information with the United States to assess the risks their nationals pose, or they present other heightened risk factors," Francisco wrote. "Whereas prior orders of the President were designed to facilitate the review, the [current] Proclamation directly responds to the completed review and its specific findings of deficiencies in particular countries."
Hawaii argued that the current ban is worse than the previous ones, because it is permanent unless the administration takes some action to amend it.
"The President has issued a proclamation, without precedent in this Nation's history, that purports to ban over 150 million aliens from this country based on nationality alone," said Hawaii's brief to the court, written by Washington lawyer Neal K. Katyal. "The immigration laws do not grant the President this power: Congress has delegated him only a measure of its authority to exclude harmful aliens or respond to exigencies, and it has expressly prohibited discrimination based on nationality."
The case is Trump v. Hawaii.