Bottles and other missiles were thrown at police as thousands of Celtic fans celebrated the club's fifth successive Scottish Premiership title win on the streets of Glasgow.
Police said about 3,000 supporters gathered in the Trongate area of the city's east end after a 3-1 win at Parkhead stadium dashed Heart of Midlothian's dream of winning their first top flight league trophy in 66 years.
Images from the scene show lines of police with riot shields and one Celtic supporter with blood streaming from a head wound.
Police Scotland said they faced "significant levels of disorder" during which officers were injured.
They said they were met with "unacceptable violence and hostility" when they went into the celebrating crowd to assist with a medical emergency.
Police said specially-trained public order officers were deployed and action was taken to restore order, disperse the crowd and make arrests.
They said extensive enquiries were under way to identify others involved and make further arrests.
Police said a significant amount of alcohol and pyrotechnics were seized by officers across the day.
Assistant Chief Constable Mark Sutherland said that while the majority of fans respected the appeal to stay away from the area, several thousand still descended on the Trongate area, some of whom engaged in serious disorder.
"Once again, we have seen unacceptable levels of violence and anti-social behaviour and Police Scotland's officers have been left to deal with the consequences," he said.
"Being assaulted is not part of the job and regrettably a number of them were injured.
"The levels of criminal behaviour witnessed, in particular the hostility directed towards our officers, is simply unacceptable and cannot be tolerated.
"Those involved should be in no doubt that they will be identified and arrested."
The match itself was one of the most highly anticipated clashes in recent Scottish footballing history.
Hearts only needed a draw to end the 40-year dominance of the Glasgow clubs but two quick goals in the final minutes secured Celtic the win they required.
Ecstatic Celtic fans poured onto the pitch as the third goal went into the net.
BBC Sportscene pundit Allan Preston said some of the Celtic fans appeared to be targeting Hearts captain Lawrence Shankland.
The match, which was approaching the end of the indicated eight minutes of added time, did not restart after the pitch invasion.
The Hearts players and staff made a hasty exit from the ground, with some players still in their match kit on the team bus as they headed back to Edinburgh.
Asked about reports that fans had assaulted Hearts players, Celtic boss Martin O'Neill said: "If some Hearts players have been accosted, then that's just not right at all.
"I genuinely didn't see the scenes at the end because I thought the game wasn't over, so I'm trying to push some of our fans off the pitch again to play whatever's left."
Hearts later released a statement saying: "Heart of Midlothian utterly condemns the shameful scenes at Celtic Park this afternoon which have, once again, embarrassed Scottish football."
"Reports of serious physical and verbal abuse towards our players and staff, both on the pitch and elsewhere, are deeply disturbing.
"We are investigating this fully and are in dialogue with Police Scotland."
The statement said that, given the menacing and threatening atmosphere inside the stadium, Hearts entire staff had no alternative but to leave immediately, without undertaking post-match media duties.
"The pitch invasion caused a chaotic end and nobody seemed to know whether or not the match had been brought to a finish," it said.
"Our players were then denied the opportunity to thank our magnificent fans – sensational to a person – for their backing this afternoon and all season long.
"We expect the strongest action possible to be taken by the footballing authorities in the interests of protecting the safety of players and supporters, and the integrity of our game."
Earlier, as his team lifted the trophy, O'Neill, now aged 74, said the Celtic players and coaching staff had "given me a reason to live".
Celtic captain Callum McGregor praised the mentality of his players after everyone "wrote us off".
"We knew we would get a chance for sure, this is the way this group is, they just keep going and going," he said.
Many Celtic fans watched the game in bars across Glasgow, among them Ian who followed the match in Connolly's.
"That was exceptional - St Martin of O'Neill delivered for his Celtic family." he said.
Another Celtic fan, Aidan, said the result had topped the birth of his two children.
"I've never experienced anything like this in my life," he said.
Fewer than 800 Hearts fans were inside the stadium in Glasgow but thousands had gathered in pubs in Edinburgh to watch on TV, hopeful that their side would pull off a remarkable footballing achievement.
Fiona, who was at the Tynecastle Arms with her 10-year-old daughter Olivia, said she was "proud and devastated" following the result.
"A lot of my friends think it's 11 men kicking a ball around a field but it's more than that for us," she said.
She added that there was "so much to look forward to" and she was hopeful for the future.
"Whether it's next year or whether it's five or ten years - it will happen eventually," she said.
Many fans gathered at Tynecastle stadium to welcome home the players, despite the disappointing result.
Edinburgh's Lord Provost Robert Aldridge sent his commiserations to the club, saying it was a heart breaking ending for a team that had given "absolutely everything" at Celtic Park.
"The club has had such a great campaign, bringing so much joy, excitement and inspiration to their fans and to the wider community," he said.
There was a heavy police presence in Glasgow both before and after the match.
O'Neill, who last won the title in 2004 during his first spell at Celtic, was among those had urged fans to celebrate "safely and responsibly".
In recent years some Celtic fans have been involved in anti-social behaviour after gathering in their thousands in the Merchant City and Trongate areas of Glasgow.
As a precautionary measure glass was removed from bus stop shelters on Friday, and the club urged supporters to avoid mass gatherings which have a "high impact" on the city.
