Two police officers were injured as police contained demonstrators on Whitehall.
There have been occupations in at least 10 universities, including at Oxford University's Bodleian Library.
School pupils walked out of lessons to join university and college students on local protest marches across the UK.
As darkness has fallen, fires have been started and windows broken in Whitehall by demonstrators who are being contained by police.
Earlier a police van was attacked and barricades thrown as protesters tried to break through police lines.
There have been reports of 15 arrests and 11 members of the public are believed to have been injured.
The Prime Minister's spokesman said: "People have a right to engage in lawful and peaceful protest, but there is no place for violence or intimidation".
There is plenty of psychological research that has demonstrated that people ("normies" at least) lose all sense of self in a sufficiently large crown. Fear, hatred and anger spread like wildfire. This is why mass-panic sets in for no reason and people are trampled to death, why protests inevitably reduce to violence.
You cannot consider a gathering of people as human beings. They must be treated like all animals: if they turn rabid, put them down.