The Scottish Football Association has said it is "appalled" by scenes of disorder after Hibs' historic victory over Rangers in the Scottish Cup final.
Fans spilled on to the pitch after the Edinburgh side beat the Ibrox club 3-2 with a late goal in injury time.
Rangers said some of their players and staff were assaulted. The club has condemned what it described as the "disgraceful behaviour of Hibs fans".
The SFA said a full investigation into the crowd trouble would be carried out.
Police Scotland described the behaviour of some fans as "totally unacceptable and inexcusable".
The force said a dedicated investigation team was being set up to identify those involved in acts of disorder and violence.
Hibs chairman Rod Petrie told BBC Scotland what happened was "deeply disappointing".
A late header from David Gray saw Hibernian end their 114-year wait to lift the Scottish Cup.
Anthony Stokes scored twice for Hibs, with Rangers goals from Kenny Miller and Andy Halliday.
Scottish Football Association chief executive Stewart Regan said: "Scottish FA is appalled by the scenes of disorder on the field immediately after full time and at the contemptible behaviour that ensued.
"What should have been an historic, memorable Scottish Cup final will now, sadly, be remembered for all the wrong reasons.
"The Scottish FA, along with Police Scotland and the stadium management have initiated a full investigation into the scenes that unfolded to understand how such a volume of supporters were able to enter the field."
Rangers said some of their players and staff had been assaulted by Hibs fans as they tried to leave the pitch. The team received their runners-up medals in the dressing room.
As Hibs fans returned to the stands part of the pitch was ripped up and the goal posts were broken by people climbing on them.
The invasion delayed the presentation of the trophy and there was no lap of honour by the Hibernian players.
'Significant safety issue'
A statement from Rangers condemned the "disgraceful behaviour of Hibs fans".
A spokesman said: "Rangers players and staff were assaulted by these fans who invaded the pitch and in the interests of their safety could not return to the pitch for the medal presentation.
"There can be no place for the violent behaviour witnessed at the end of the final and Rangers fully expect the Scottish Football Association and Police Scotland to launch an investigation to find out where security measures failed.
"The club would like to commend Rangers fans for the restraint they showed under severe provocation."
In a post match interview with BBC Scotland, Hibs chairman Mr Petrie said: "It is deeply disappointing that this happened. It shouldn't have happened.
"The images should have been about the club celebrating the win as opposed to any other aspect of it.
"The fact that it has happened, then you have to look at what the response was to it and the response seemed to be entirely professional, to contain the incursions on to the pitch and eventually to put people back into their seats."
Ch Supt Kenny MacDonald of Police Scotland said: "Officers, including the force's mounted branch, responded and brought the matter under control as quickly as possible.
"We understand that feelings run high at these events and fans want to celebrate their team's win, however the reckless behaviour of fans entering the pitch is totally unacceptable and inexcusable. This created a significant safety issue.
"This action led to an outbreak of disorder between rival fans on the pitch. A dedicated investigation team is being set up to identify those who have carried out these acts of disorder and violence."
Police horses were last needed at the 1980 Scottish Cup final between Celtic and Rangers.