Brazilian President Dilma Rousseff 'proud' of protests
Brazilian President Dilma Rousseff has said she is proud of the tens of thousands of people who have taken to the streets to demand better education, schools and transport.
"My government is listening to the voices calling for change," said Ms Rousseff in her first comments since Monday night's demonstrations.
Some 10,000 people gathered in Sao Paulo on Tuesday for a new protest.
Scuffles broke out with police outside the office of Mayor Fernando Haddad.
Police officers sought refuge inside the building, while demonstrators trying to stop the attack clashed with groups of masked activists.
Stones were thrown and windows were broken.
Journalists were attacked and a car set alight, while protesters set fire to a police station in another part of the city, says the BBC's Luis Kawaguti.
Demonstrations were also taking place in the states of Rio de Janeiro and Minas Gerais.
The protests began with demands for bus fare hikes to be revoked.
They have turned into a nationwide demonstration against bad governance.
"Brazil has woken up a stronger country," said President Rousseff.
"The size of yesterday's marches is evidence of the strength of our democracy."
"It is good to see so many young people, and adults - the grandson, the father and the grandfather - together holding the Brazilian flag, singing our anthem and fighting for a better country," said Ms Rousseff.
She said her government had lifted "40 million people into the middle class" but more needed to be done to improve access to free health and education.
The demonstrations are Brazil's largest since 1992, when people took to the streets to demand the impeachment of President Fernando Collor de Mello.
The Sao Paulo mayor was earlier said to be heading for a meeting with President Rousseff, apparently to discuss options that would enable him to meet protesters' demands and cut public transport fares, says the BBC's Gary Duffy.
The mayors of Cuiaba, Recife, Joao Pessoa and other cities have announced a reduction in bus fares in response to Monday's protests.
A major rally is set to take place in Rio on Thursday.
The current wave of protests began earlier this month, with marches in Sao Paulo against a hike in the price of bus fares, from 3 reals ($1.40, £0.90) to 3.20.
They have been organised largely by young people through social media. The organisers called the movement Passe Livre (or Free Access).
Monday night saw the biggest demonstrations since the movement began.
In Brazil's largest city, Sao Paulo, about 65,000 people took to the streets.
The largest march was in Rio de Janeiro, where some 100,000 people marched peacefully through the city centre.
Towards the end of the evening in Rio, there were violent clashes between groups of protesters and police.
The Rio de Janeiro state assembly was attacked, shops were vandalised and a car was set alight.
There were also isolated incidents in Sao Paulo, Brasilia, Porto Alegre and other cities.
"The isolated acts of violence, carried out by a minority, should be vigorously condemned," said Ms Rousseff.
The demonstrations gathered pace as Brazil hosts the Confederations Cup, a curtain-raiser event for next year's football World Cup.
Many complained of the huge amounts spent on construction for the World Cup and the 2016 Olympics, which will be hosted by Rio de Janeiro.
"We need better education, hospitals and security, not billions spent on the World Cup," said one mother who attended the Sao Paulo march with her daughter.
Deputy Sports Minister Luis Fernandes said the World Cup had provided an opportunity for the country to invest in infrastructure which will benefit everyone.
"There is widespread support in the Brazilian population for the World Cup and the Confederations Cup," said Mr Fernandes.