The PM's chief adviser told a press conference he wanted to "clear up confusion" amid calls for him to resign.
SNP: A car-crash press conference
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Reacting to Dominic Cummings statement, the Scottish National Party's Westminster leader Ian Blackford said: "After that car-crash press conference Boris Johnson has no option but to sack his chief adviser.
"It is now beyond doubt that Dominic Cummings broke multiple lockdown rules, possibly on multiple occasions."
Labour: One rule for Cummings, another for the rest
Knowledge of Cummings' trip show Johnson 'not fit' to be PM
Drivetime with John Beattie
BBC Radio Scotland
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Alastair Campbell has suggested Cummings' statement demonstrates Boris Johnson is "not fit" to be prime minister.
He said: "He's known that his closest adviser broke the rules while they've been pumping this message into our house day after day after day after day: stay at home, protect the NHS, save people's lives."
Tony Blair's ex-spin doctor said Mr Cummings had "no idea how angry people are". He added: "We've got a depraved, amoral government and he's a big part of it, and they're going to try tough it out."
Cummings defends position, but still faces problems...
Dominic Cummings should concede he made a mistake rather than retrospectively rewrite the rules, says Sturgeon.
Analysis: 'What have we learned from Cummings' side of the story?'
BBC political correspondent
Dominic Cummings has given a detailed account of what he did, when and why. So what have we learned from his side of the story?
He described the fact that his London home had become a “target” which led him to fear for the safety of his family.
Mr Cummings also admitted not telling the prime minister about his decision to decide to travel to his parent’s property in Durham.
He explained some of the uncertainties about his movements including what he was doing in Castle Bernard (to test his eyesight for driving) and whether he stopped on the journey from London (he didn’t).
But on several occasions Mr Cummings described the “exceptional circumstances” of providing care for a small child, which he believed the guidelines allow.
He acknowledged that people were angry and “hated the idea of unfairness” - and admitted that he should’ve made a statement sooner.
But this was an explanation for his actions, not an apology.
It will be for people to judge whether they accept it as a justification for what many see as acting against the spirit, if not the letter of the rules.
Cummings: I don't think I am so different
Sky's Beth Rigby asks why Dominic Cummings thinks he is so different to "thousands of ordinary families" and does he owe them apology.
"I don't think I am so different," he says.
"I think that I looked at the guidance, I knew what it was, it talks about exceptional circumstances with small children... and I think I behaved responsibly and legally."
Laura Kuenssberg asks if he has exploited a "loophole" in the rules and if he understood why people were angry.
"I certainly do," Mr Cummings replied. "After some of the media over the last couple of days I am not surprised you are very angry if you are someone sitting at home watching the media.
"But I think and hope that today explained all the circumstances about a complicated tricky situation."
He says it may have been a "mistake" not to tell the PM on the Friday night when he decided to leave, adding he has "thought a lot over this period what I could have done better in dealing with the whole crisis".
"There are lot of things I could have done better but in this 14 days [my actions were] reasonable."