"It was on Highway 1 by the Cambodian border," he told PEOPLE recently. When he took the photo, Ut was a young photographer with the Associated Press, covering the war that was destroying his native country.
On the morning of June 8, Ut saw a group of refugees traveling down the highway. The South Vietnamese army had been fighting the Viet Cong outside the villages there, and the people living in the area were forced to flee.
Ut was on the scene when South Vietnamese planes, thinking the refugees were Viet Cong, started bombing them.
"There were a lot of people dead – a lot of children," he recalls. He was about to turn back when he saw the final plane drop its bombs.
Instantly, he knew: "I said, 'Oh my God, the napalm.' They had bombed all morning, but not with napalm. Never in my life have I seen what I saw."
Nick Ut / AP
Through the smoke, he saw something else: A young girl, naked, running screaming down the road.
"I thought, 'Why doesn't she have clothes?' Ut recalled. He ran towards her and snapped a photo. Then he put the camera away.
Nick Ut / AP
With the help of a friend from Britain's ITV, Ut covered the girl, 9-year-old Kim Phuc, in a coat and took her to the hospital, alongside other children wounded in the bombing. In the 40-minute drive to an American hospital, Phuc kept repeating, "I think I'm dying."
Once they arrived, Ut used his media pass to convince the hospital staff to admit the children. It was the end of a day that would link them for the rest of their lives.
"After my brother died, the AP was my family now," Ut recalled. "I wanted to replace my brother."
Before his death, Ut's brother had told him he hoped he could help end the war by photographing the horrors of combat. Ut's brother was the seventh child in their family, and Ut remembers thinking of him on that highway.
Richard Vogel / AP
"My brother gave me that picture," Ut says now. "Good for him."
Na Son Nguyen / AP
It was almost never seen at all – cautious journalists in America wondered if the nudity was too extreme for their front pages. Ut's bosses at the AP swayed them, and soon the image became the instant shorthand for the horrors of war.
Ut won the Pulitzer for it, the youngest photographer ever to do so. But if the picture made Ut a celebrity (at least in the world of photojournalism) it made Phuc a symbol.
Robinson / AP
Eventually, though, Phuc's fame paid off. The government allowed her to go to school in Cuba, where she fell in love with another Vietnamese student. In 1992, coming back from their honeymoon, the newly married couple sought asylum in Canada. Today Phuc is a United Nations Goodwill Ambassador living in Ontario with her husband and their two sons.
Jim Caccavo / AP
"I can accept the picture as a powerful gift," Phuc told the AP in 2012. "Then it is my choice."
Ut, for his part, looks at the picture every day; it's in his room in his house in Los Angeles, where he moved in 1977 after the fall of Saigon.
"Every time I see it I cry a little bit inside," he explained, "because Kim was hurt so badly. She is like my family."
Now, 48 years after he started working for the AP, Ut works the agency's L.A. beat. He regularly returns to Vietnam to see Phuc's family and give money to the children there, but he says he's had enough of shooting war.
"When you're over 60, what do you do? I have children," he explained. "I'm fighting the paparazzi in Hollywood – that's my war."
Jae C. Hong / AP