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.
Home Secretary Priti Patel described the violence as “thoroughly unacceptable thuggery”.
Some anti-racism demonstrations have also taken place across the UK, including in central London.
The Met Police had placed restrictions on several groups intending to protest, including having to finish at 17:00 BST, following violent scenes last weekend.
However, several groups remained on the streets of central London after the cut-off.
As some protestors apparently moved towards Waterloo Station around 18:00 BST, both the underground and mainline station were temporarily closed due to the protests – but later reopened.
Mayor of London Sadiq Khan urged demonstrators to “leave now” to prevent further violence and the spread of coronavirus in the city.
Various groups from around the country, including some far-right activists, said they had come to London to protect symbols of British history.
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 right-wing protesters then moved to Trafalgar Square, where fireworks were thrown across the crowds.
A statement from the London Ambulance Service said it had treated 15 patients, including two police officers, for injuries at the protests.
“Six of these patients – all members of the public – have now been taken to hospital.” The ambulance crews remain at the scene.
Police attempted to stop protesters getting to Hyde Park where an 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
From mid-morning hundreds of men – most of whom were already drinking – headed into Parliament Square.
Some were members of football hooligan networks who had put aside their differences for the day. Others were from the far-right.
But despite claims that the day was about defending British heritage, it soon became violent.
Groups looked for opportunities to attack the police. Bottles and cans were thrown at their lines and horses – smoke bombs and fireworks set off. Journalists who came too close were threatened.
One bare-chested man kicked a police officer to the ground and there were repeated attempts by others to storm barriers as the police contained the trouble.
By late afternoon, some had left to find more beer – only to find their way back blocked by riot police.
Others tried to regroup on nearby streets as the police played cat and mouse – using their numbers to repeatedly contain and block attempts to take violence around the city.
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.
The Met Police have imposed measures requiring all demonstrations to come to an end at 17:00 BST and giving officers enhanced powers to stop and search individuals, after police learned some people were coming into London to cause harm and were likely to bring weapons with them.
The Section 60 order will remain in place until 02:00 BST on Sunday.
The 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.