Flares and smoke bombs have been thrown in Trafalgar Square
Police were pelted with bottles during confrontations with demonstrators in London, where hundreds gathered despite warnings to avoid protests.
Groups gathered in the centre of the capital, claiming they were protecting statues from anti-racism activists.
Some anti-racism demonstrations have also taken place around the country, including in central London.
The Met Police has placed restrictions on several groups intending to protest, following violent scenes last weekend.
Various groups from around the country, including right-wing activists, said they had come to London to protect symbols of British history.
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Hundreds of mostly white men gathered around the Cenotaph war memorial in Whitehall and the boarded-up statue of Winston Churchill in Parliament Square.
There were a number of clashes with police in riot gear as crowds chanting “England” and raising their arms surged towards lines of officers.
Some protesters managed to break metal barriers around the Cenotaph on Whitehall while hurling flag poles, a smoke flare and a traffic cone towards police who were striking them back with batons.
Large groups of far-right protesters then moved to Trafalgar Square, where fireworks were thrown across the crowds.
Police attempted to stop them getting to Hyde Park where a anti-racist demonstration, which had largely been peaceful, was taking place.
Organisers from the Black Lives Matter movement had urged people not to join any anti-racism rallies planned for the weekend over fears there could be clashes with far-right groups. One demonstration planned for Saturday in London was brought forward by a day.
Sharing footage of the clashes with police on Twitter, Home Secretary Priti Patel described it as “unacceptable thuggery”.
“Any perpetrators of violence or vandalism should expect to face the full force of the law,” she wrote.
“Violence towards our police officers will not be tolerated.”
The Met Police Federation described it as “unacceptable”, tweeting that its officers “do not come to work to face this level of violence and abuse”.
At the scene in Parliament Square
By BBC home affairs correspondent Dominic Casciani
Shortly after 13:00 BST a black woman, wearing a mask, was spoken to by police as she was entering Parliament Square from near the Supreme Court.
The officers and the woman were quickly surrounded by a jeering crowd. When officers asked the woman to get down from a plinth, one of the protesters appeared to try to slap the woman.
Part of the crowd surged as additional police officers in riot gear were brought into the scene with horses to strengthen their lines.
The crowd then threw bottles and cans at the officers and let off a number of smoke bombs. One officer appeared to be pushed to the ground as he was taking the woman away.
Speaking before the clashes, the leader of the far-right group Britain First, Paul Golding – convicted last month of an offence under the Terrorism Act – said they had turned out to “guard our monuments”.
The statue of Churchill was boxed up to protect it from potential damage, after protesters daubed “was a racist” on it last weekend.
Hundreds of people also gathered in Glasgow, Bristol and Belfast as part of events organised to “protect” war memorials.
In Brighton, some anti-racism protesters clashed with right-wing groups gathered at the city’s War Memorial.
That came after more than 1,000 protesters had gathered, wearing black clothes and masks, to form a mile-long line along the seafront as they held a silent protest.
In Newcastle, demonstrators supporting the Black Lives Matter movement were heckled by a large counter-protest.
Denise Richards, who is involved in the Black Lives Matter movement in Derbyshire, said her chapter had decided not to protest in London on Saturday.
She told BBC Radio 4’s Today programme that peaceful protesters feared they would be caught up in violent clashes with far-right demonstrators and this could “tarnish” the work of Black Lives Matter.
Nick Lowles, chief executive of campaign group Hope Not Hate, said there was a “very serious” threat of trouble from far-right activists and commended Black Lives Matter for standing down their plans to protest in London on Saturday.
“There are some people who are genuinely concerned about the protection of their statues and monuments but many people are coming for a fight and they are talking openly about it on their social media accounts,” he told the programme.
A Black Lives Matter demonstration took place in central London on Friday evening with leaders of the march urging those in attendance to keep the demonstration “peaceful” and not to join any anti-racism rallies planned for the weekend.
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Mayor of London Sadiq Khan urged people to stay away from central London on Saturday, saying there was a risk of violence and disorder from extreme far-right groups planning to travel to the capital.
The Met Police said it had put a Section 60 order in place until 02:00 BST on Sunday after it learned some people were coming into London to cause harm and were likely to bring weapons with them.
The measures require the demonstrations being held to end at 17:00 BST on Saturday, as well as giving officers enhanced powers to stop and search individuals.
The police restrictions come in the wake of violence and serious disorder in Westminster at the end of protests last weekend.
Demonstrations have been taking place across the world following the death in police custody of African American George Floyd in Minneapolis last month.