Irish police confirmed a total of five people were injured following the 'major security' incident which occurred shortly after 1.30pm on Parnell Square.
One of the three children, a girl, and a woman in her 30s suffered 'serious' injuries, and a man in his 50s and two other children were also among the injured, police said. All five were taken to hospital.
Police said the man in his 50s, who is receiving treatment for his injuries in Dublin, is currently 'a person of interest' and they are not looking for any other person at this time.
Passersby overpowered the assailant outside a school and were able to disarm him, witnesses have said.
Taoiseach Leo Varadkar confirmed Gardai have detained a suspect following the incident in Dublin city centre.
'The kids were out walking. All of a sudden one of them fell to the ground, then another fell to the ground, then another falls to the ground,' a man who saw the attack unfold told the Irish Independent.
'Then this guy started running past', he said, and 'a load of people jumped on him.'
Following the attacks, a furious row broke on Parnell Street near the site of the stabbings, with a man being accused of 'defending' the attacks, with several people outraged by the incident that occurred near a school.
Speaking at a press conference in Dublin, Superintendent Liam Geraghty said An Garda Siochana 'continues to have an open mind at this early stage of the investigation', but are satisfied there is no terrorist link.
'This is a very early stage of the investigation which is not even four hours old at this stage, but at this time An Garda Siochana is satisfied from our inquiries that there is no terror-related activity or related to any wider aspects in relation to this matter,' he said.
'It would appear to be a standalone attack, and we need to determine the reasons behind that.'
Superintendent Geraghty said his understanding is that a knife was used in the attack, but he could not provide more detail on the nature of the injuries.
He added: 'An Garda Siochana continue to investigate all of the circumstances of a serious assault which occurred on Parnell Square East shortly after 1.30pm this afternoon.
'Preliminary indications are that a male attacked a number of people on Parnell Square East.
Comment: Reports from on the ground allege that the perpetrator is an Algerian migrant:
'Five casualties have been taken to hospitals in the Dublin region.
'These casualties include three young children, an adult female and an adult male.
'One girl, aged five years, has sustained serious injuries and is currently receiving emergency medical treatment in CHI Temple Street.'
One of the three children, a girl, and the woman sustained 'serious' injuries in the incident on the square found in the north inner-city of the Irish capital.
The man and the two other children sustained less serious injuries, the Gardai said, which added it had been in contact with the parents of the injured children.
Police said they are following a 'definite line of inquiry' and that they were not looking for any other person at this time.
A large knife has been seized from the scene, local reports said.
The incident took place outside Gaelscoil Coláiste Mhuire, a school for primary-aged children. Police are working to establish a motive but do not suspect a terrorist motivation behind the attack, Irish national broadcaster RTE reported.
Several people intervened and disarmed the man, witnesses said.
One said they saw a large knife being taken from the suspect and thrown under trees opposite Rotunda Hospital, which sits in the middle of the square.
Another told how she and another bystander disarmed a man with a knife.
'Without thinking, I just took across the road to help out,' Siobhan Kearney said, describing the scene as 'absolutely bedlam' to RTE.
'Two children and the woman were taken back into the school where they were coming from' as the man was restrained on the ground by several people.
'Me and an American lady formed a ring around him saying we'd wait on the Garda.'
Kearney said the children were about six years of age, and that she believed the woman who was injured was either a teacher or a bystander.
Other eyewitnesses told the Irish Daily Mail that they saw a young girl 'around four years of age' receiving emergency treatment from a passerby.
He said: 'I saw a young girl with blood all over her and a woman was giving her treatment. There was just chaos everywhere. Everybody was in tears.
'I can't imagine what the parents are going through.'
Dublin city councillor Nial Ring said he had spoken to a woman who was on her way to the Stardust inquiry.
He said the woman 'on seeing the attack confronted the attacker with another man (I understand he was a Deliveroo cyclist) and managed to get the knife off him.
'Those two citizens are real heroes and probably prevented even more carnage.'
'Five casualties have been taken to various hospitals in the Dublin region,' a Garda spokesperson said.
'The casualties include an adult male, an adult female and three young children.
'One child, a girl, has sustained serious injuries, the other two children are being treated for less serious injuries.
'An Garda Siochana is in contact with parents of all three injured children.'
Superintendent Liam Geraghty said a five-year-old boy and six-year-old girl were treated for less serious injuries after the incident in Dublin.
He said: 'The boy has since been discharged from CHI Crumlin.
'An Garda Siochana is providing support to the parents of all the children.
'The adult female in her 30s is being treated for serious injuries at the Mater Hospital.
'An adult male in his 50s is also being treated for serious injuries at a hospital in the Dublin region.
'The scene remains sealed off at this time and a technical examination of the scene is ongoing.
'An incident room has been established at Mountjoy Garda Station and a senior investigating officer has been appointed.
'I have been in contact with the Minister for Justice who is keeping me updated. The facts in this matter are still emerging. 'The emergency services responded very quickly and were on site within minutes. I thank them for that.
It is understood that at the same time the attack was unfolding, an inquest was being heard nearby on the Stardust nightclub tragedy, a fatal fire that tore through the Stardust nightclub in Artane, Dublin in 1981, killing 48 people.
A witness from the inquest, who also witnessed the attack, said a number of young men intervened to help as today's attack unfolded.
They said emergency services arrived at the scene within minutes.