On the tiny Italian island of Lampedusa, the Rev. Stefano Nastasi threw the ecclesiastic equivalent of a Hail Mary pass.

Legions of largely Muslim refugees looking for a better life in Europe were reaching the island from North Africa only to perish, be turned back or be sent to languish in camps. Troubled by their plight, the priest dispatched a letter to the Vatican: Would Pope Francis come and highlight the humanitarian crisis in his new backyard?

When the answer finally came, Nastasi said he was as surprised as anyone: Why yes, in fact, the Holy Father would. In something of a trial run ahead of the first major trip of his papacy — a weeklong visit to Brazil from Monday — Francis touched down on the island July 8 and promptly acted in a way that got observers buzzing about the rise of a revolutionary pope.

Only days earlier, Mercedes-Benz had presented Francis with a new bulletproof “Popemobile.” But the first Latin American pontiff, who has largely rejected the lavish trappings of his office, traveled around the island in a borrowed, open-top Fiat.

His predecessor’s words once sparked riots in predominantly Muslim countries. But from an impromptu altar made out of a rickety raft, Francis welcomed “the dear Muslim immigrants who are beginning the fast of Ramadan.” He called for more humane treatment of immigrants regardless of faith and an end to the “globalization of indifference.”

“Who has wept for the deaths of these brothers and sisters?” he asked, referencing seven refugees who recently drowned after fishermen failed to go to their aid near the island. “Who has wept for the people who were on the boat? For the young mothers carrying their babies? For these men who wanted something to support their families?”

In the four months since he became pontiff, Francis has emerged as the great hope of a church facing a management crisis at its headquarters and major challenges from secularism and rival faiths.

Talk now is less about the irreversible decline of an out-of-touch papacy and more about whether Francis will fulfill the early hype and connect with a complex, diverse and increasingly modern global audience in a way that his predecessor, Pope Benedict XVI, never really could.

Francis’s biggest test yet comes this week, during his visit to Brazil — the world’s largest predominantly Roman Catholic nation. There, he will find a microcosm of the church’s greatest opportunities and most thorny challenges.

The percentage of Catholics in Brazil dropped from 93 percent of the population in 1960 to 65 percent in 2010. But the new pope has piqued the interest of Brazilians such as Paula Mora, 18, a university economics student. The daughter of practicing Catholics, she grew up distant from the church and never attends Mass. And yet, she said, she can’t help but notice how Francis has rejected the luxuries associated with the church’s hierarchy and gone out of his way to bond with ordinary people.

“I think it’s exciting — it is a big deal,” she said of his trip. “I have a lot of sympathy with this pope, more than with the others.”

In a sign of the “Francis boom,” the new pontiff’s Sunday blessings in St. Peter’s Square are drawing far greater numbers than Benedict’s ever did. In Italy, a recent poll put his approval rating at 92 percent, with 13 percent saying his arrival had led them to attend Mass more frequently.

In the United States, an April poll by the Pew Research Center showed that 84 percent of Catholics held a favorable view of the new pope, compared with 67 percent who felt the same about Benedict in the first year of his papacy.

Francis’s image is transcending the spiritual world of youth group sing-alongs and Sunday Mass. Atheists are penning odes to his walk-the-walk manner. The Italian edition of Vanity Fair named him Man of the Year. Suzy Menkes, longtime style editor of the International Herald Tribune, cited the pope’s aesthetic of austerity for a shift in Milan fashion toward “discreet, sober, even chaste outfits.”

CNN recently pondered whether the compassionate, media-savvy new pope is the modern-day Diana, Princess of Wales.

“I would sum up the way American Catholics are feeling with one word: finally!” said John Thavis, author of “The Vatican Diaries.” “Finally, we have a pope who is a real human being and who understands that you can’t teach poverty while you’re sitting on a gilded throne.”

Still, experts say Francis’s impact will remain checked until Catholics see how he handles issues that have truly divided the church, including the role of women.

One of the most energizing events in years for many in the U.S. church was last year’s “Nuns on the Bus” campaign, during which a group of sisters toured the country to protest proposed government budget cuts for the poor. Many Catholics were angered by the Vatican’s move to investigate the nuns for focusing too little on such issues as abortion.

Weeks after taking office, Francis chose to reaffirm the investigation, alarming even more liberal Catholics.

This month, he updated Vatican laws to explicitly criminalize abuse and possession of child pornography. But critics largely dismissed the effort as token. They call the Vatican’s still unclear response to a new request from a U.N. panel for information on church sex abuse cases a better test of how much transparency Francis will truly usher in.

And yet Francis, some argue, has already begun a process that could shake up business as unusual in the Vatican City. For instance, he has continued in the same austere vein as he did while cardinal of Buenos Aires, declining to live in the 10-room papal apartment and instead moving into quarters typically reserved for Vatican guests.

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