50 House Democrats boycott inauguration
Hillary Clinton, who lost to Mr Trump in a dramatic upset in November’s election, was in attendance with her husband, former President Bill Clinton.
President Barack Obama shook hands with each member of the Trump family before the ceremony.
Former presidents and first ladies, including George W Bush and his wife Laura as well as Jimmy Carter, were in attendance.
The only absences were 92year-old George Bush Senior, who is in hospital being treated for respiratory problems, and his wife Barbara.
Members of Congress were also in attendance, although more than 50 House Democrats had refused to attend the ceremony.
The 45th president has vowed to roll back many of his predecessor’s policies, including repealing Mr Obama’s signature health care law and building a wall along the US-Mexico border.
Earlier, about 150 protesters dressed in black marched through Washington, smashing windows and rolling rubbish bins into the street to form blockades.
Thousands of police have been deployed and many kilometres of crowd barriers set up as part of a vast security operation ahead of the inauguration.
Some spectators had waited for hours.
“I’m here for history,” said 24year-old Kevin Puchalski. “This is the first president that I voted for that won.”
Chris Lehmann, a 55-year-old maintenance supervisor from Belmar, New Jersey, said: “I’m so excited, I’m like, on top of the world.”
An estimated 800,000 to 900,000 people travelled to Washington, although it is not clear how many are coming to celebrate the inauguration and how many to protest against it.
Demonstrations for and against Mr Trump include a rally for the new president by Bikers for Trump after the ceremony.
The Women’s March on Washington on Saturday - for racial and gender equality, and other issues perceived to be under threat from Mr Trump’s administration - is expected to draw about 200,000 people.
Meanwhile in New York, thousands of people attended a rally where dozens of celebrities and politicians voiced their concerns about the president-elect.