Bolsonaro was sworn in as president on Tuesday, promising to overhaul the country's economy and bring about sweeping social change.
An admirer of US President Donald Trump, Bolsonaro rose to power on a pro-gun, anti-corruption agenda.
Speaking in an address to the nation on Tuesday, Bolsonaro said now that he had taken power, his country had been "liberated from socialism and political correctness."
In an earlier inauguration speech, the 63-year-old former paratrooper had promised to "unite the people, value the family, respect religion and our Judeo-Christian tradition, combat the ideology of gender and preserve our values."
Bolsonaro, who had previously said his education minister would stop "Marxist trash" from being taught in schools and universities, vowed his government would "prepare children for the job market, not political militancy."
Bolsonaro, a former congressman whose views were once considered to be at the fringe of mainstream politics, has promised to end policies that protect criminals. He has called for police to be shielded from prosecution for shooting criminals, and for gun laws to be eased for "good citizens" to counter armed criminals.
"I call on all congressmen to help me rescue Brazil from corruption, criminality and ideological submission," he said.
Also inaugurated in the event before Brazil's Congress in the capital, Brasilia, was Bolsonaro's vice president, Hamilton Mourao, a retired general. Both rose to power capitalizing on dissatisfaction that followed four presidential wins by the left-wing Workers' Party.
Right-wing foreign leaders attending the event included Israeli Prime Minister Benjamin Netanyahu and Hungarian Prime Minister Viktor Orban. Leftist presidents such as Nicolas Maduro of Venezuela, Daniel Ortega of Nicaragua and Miguel Diaz-Canel of Cuba – whom Bolsonaro has deemed dictators – were not invited.
The US was represented by Secretary of State Mike Pompeo, who met Netanyahu on the sidelines of the event.
Ahead of the ceremony, Bolsonaro and his wife, Michelle, traveled through the city by motorcade, standing in an open-top Rolls-Royce surrounded by onlookers. Supporters chanted "Brazil above everything, God above everyone."
The pair were surrounded by dozens of guards on horseback and plain-clothes bodyguards who ran alongside them.
Security was ramped up for the inauguration, with some 3,000 police patrolling the event. Bolsonaro was stabbed in the stomach in a September attack by a knife-wielding lone assailant.
Read more: Is Brazil turning into a military state?
US President Donald Trump congratulated his Brazilian counterpart via Twitter on a "great inauguration speech."
Journalists attending the inauguration complained about their treatment after they were required to arrive at the event seven hours early. Access to government buildings was limited and items confiscated. Bolsonaro, a vocal critic of media that voice opposition to him, has promised to cut advertising funds from state-owned companies to some media organizations.
rc/ng (AP, AFP, dpa, LUSA, Reuters)