There was no immediate word about a motive.
The student who died has been identified through social media posts as Harrison Brown, according to the Graham Independent School District. Friends and family posted messages of the loss as well.
Student Rachel Prichett said she was standing in line at a food truck outside a gym when she saw a man with a knife resembling a machete approach the person standing behind her.
"The guy was standing next to me," Prichett said. "He grabbed him by the shoulder and shoved the knife in it. I just started running as fast as I could."
Police identified the suspect as 21-year-old Kendrex J. White.
University police Chief David Carter described the weapon as a "Bowie-style" hunting knife. He said the stabbings occurred within a one-block area as the attacker "calmly walked around the plaza."
Another student, Ray Arredondo, said he was walking to his car when a mass of students near the gym started running.
"They were just screaming, 'Run! Get out of here!'" Arredondo said.
One person died at the scene near the gym. The others were taken to the hospital. There were reports of additional victims with non-life-threatening injuries, according to tweets from Travis County Emergency Medical Services.
The attacker did not resist when officers confronted him at gunpoint, Carter said.
Lindsey Clark said she saw the suspect get tackled by police as he was running toward the entrance of Jester Hall, a complex of dormitories and classrooms. She described him as wearing a bandanna and gray sweatshirt and said he appeared quiet and subdued as police held him on the ground.
"You could see and hear people running and screaming: 'There he is!'" before he was tackled by officers, Clark said.
Carter said it would be "premature" to discuss the suspect's motive and "what was going through his mind."
Arredondo later saw what looked like CPR being performed on someone outside the front door of the gym. Another student was sitting on a bench being treated for cuts to the head or neck, he said.
Authorities cordoned off the scene as a large contingent of state and local police, including officers in helicopters, swarmed the area.
The University of Texas is blocks from downtown Austin and the Texas Capitol and is one of the nation's largest universities.
White was an active member of the Black Health Professionals Organization student organization on campus, said Melody Adindu, the group's new president. She said White was passionate about his work and was "very interactive and easygoing."
Some of White's former classmates at Killeen High School, near the gates of the Fort Hood Army post in Central Texas, had similar recollections of him.
"He was a really smart guy in high school, he was always nice, had plenty of friends, and was in the International Baccalaureate program. I'm definitely surprised he would do this," Kay'Lynn Wilkerson told the Killeen Daily Herald.
Ex-classmate Angela Bonilla called White "the sweetest guy, laughing and having a good time with people".
Adindu and other students complained on social media it took too long for the campus officials to send a text alert of the attack. Texts to students showed a nearly 30-minute lag between the arrest and warning, even though city and county emergency offices were tweeting about the incident when they first responded.
Carter said White was confronted and arrested within two minutes of the first call to police and the situation was immediately under control. The campus siren wasn't used because there was no need for a lockdown to keep people in place, he said.
"There was no ongoing threat. We had him in custody as soon as we arrived," Carter said.
Governor Greg Abbott offered his prayers to everyone in Austin.
UH students react to deadly attack on UT campus
"Praying for all those affected by this heinous attack and for the UT-Austin community," Abbott tweeted.
The University of Texas - Austin student government issued the following statement: "Today, the Longhorn community mourns the unimaginable loss of one of our fellow students and the injury of three others. Our hearts are with the family, friends, and loved ones of the victims and those who witnessed the attack. This senseless tragedy has left our community shaken. We encourage Longhorns to come together and support each other during this extremely difficult time."
The attack occurred in the central campus, just a short walk from the administration building and the landmark clock tower that was the scene of a mass shooting in 1966.
"This breaks my heart that any of our students are touched by tragedy. They come here to learn, to look to the future," said university President Greg Fenves.
The stabbings came only a few days after a 19-year-old man armed with a machete wounded two people at a university coffee shop in Lexington, Kentucky.
In the April 28 attack, the assailant at Transylvania University asked about the political affiliations of people at the shop. He was arrested and charged with assault. The victims' wounds were not life-threatening, authorities said.
The Associated Press contributed to this report.
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