Video report by Political Editor Robert Peston
The government adviser has come under increasing pressure to resign but will be staying on and did not offer an apology for his actions.
Mr Cummings said during his statement in the Downing Street rose garden: “I believe in all the circumstances I behaved reasonably and legally."
Furthermore, Mr Cummings denied he had broken the “spirit” of the rules and said he had not offered his resignation to the Prime Minister.
Boris Johnson’s chief adviser said he did not speak to the Prime Minister before driving to County Durham as he answered allegations that he breached coronavirus lockdown restrictions.
Mr Cummings has faced plenty of criticism for his actions but Mr Johnson offered his support on Sunday during the daily coronavirus briefing.
"In every respect, he has acted responsibly, legally and with integrity," Mr Johnson said.
How the public reacted to Dominic Cummings's statement
Mr Cummings said his decision to drive 264 miles from London to County Durham was based not only on fears over a lack of childcare if he became incapacitated with Covid-19 but also concerns about his family’s safety.
Answering questions in Downing Street’s garden, Mr Cummings said: “I don’t regret what I did. I think reasonable people may well disagree about how I thought about what to do in the circumstances, but I think what I did was actually reasonable in these circumstances.
“The rules made clear that if you are dealing with small children that can be exceptional circumstances.
“And I think that the situation that I was in was exceptional circumstances and the way that I dealt with it was the least risk to everybody concerned if my wife and I had both been unable to look after our four-year-old.”
Mr Cummings admitted travelling to Barnard Castle by car to test if he was fit to drive the full journey back to London due to concerns over his vision after recovering from illness.
"My wife was very worried, particularly as my eyesight seemed to have been affected by the disease.
“She did not want to risk a nearly 300-mile drive with our child given how ill I had been.
“We agreed that we should go for a short drive to see if I could drive safely, we drove for roughly half an hour and ended up on the outskirts of Barnard Castle town.
“We did not visit the castle, we did not walk around the town.”
The witness who saw him at Barnard Castle, Robin Lees, told ITV News there is "nothing wrong with my eyesight," adding, "not sure driving to test your eyesight is on".
Mr Lees also said he "doesn’t remember being wished a happy Easter" by Mary Wakefield, Mr Cummings' wife, something the adviser claimed occured.
At an extraordinary press conference, Mr Cummings said stories suggested he had opposed lockdown and “did not care about many deaths”.
“The truth is that I had argued for lockdown, I did not oppose it but these stories had created a very bad atmosphere around my home, I was subjected to threats of violence, people came to my house shouting threats, there were posts on social media encouraging attacks.”
Mr Cummings said he was worried that “this situation would get worse” and “I was worried about the possibility of leaving my wife and child at home all day and often into the night while I worked in Number 10.”
“I thought the best thing to do in all the circumstances was to drive to an isolated cottage on my father’s farm.”
Mr Cummings moved to clear up the timeline of his movements from the end of March.
The adviser said he was told at around midnight on March 26 by the Prime Minister that he had tested positive for Covid-19.
After discussing the national emergency arrangements, Mr Cummings said he went to Number 10 the following day for a series of meetings.
He received a call from his wife, who was looking after their four-year-old child, who said she felt badly ill, had vomited and felt like she might pass out.
That led to Mr Cummings’ decision to swiftly leave No 10 – actions that were caught on camera in Downing Street.
After a couple of hours his wife felt better and Mr Cummings returned to Downing Street.
But he said that evening he discussed the situation with his wife – including the fact that many in Number 10 had developed coronavirus symptoms.
He was worried that if both he and his wife fell ill there was “nobody in London we could reasonably ask to look after our child and expose themselves to Covid”.
Mr Cummings said that he drove up to Durham with his wife and son and did not stop on the way.
He said the next day he woke up in pain and “clearly had Covid symptoms”.
He added: “So I drove the three of us up to Durham that night, arriving roughly midnight. I did not stop on the way.
“When I awoke the next morning, Saturday March 28, I was in pain and clearly had Covid symptoms including a headache and a serious fever.
“Clearly I could not return to work anytime soon. For a day or two we were both ill, I was in bed, my wife was ill but not ill enough to require emergency help.”
He said as he was recovering he went for a walk in the woods next to the cottage they were staying at which was private land and while they saw some people they had no interaction with anyone.
Mr Cummings said that by April 11 he was still feeling “weak and exhausted” but had no Covid symptoms so thought he would be able to return to work the following week – possibly part-time.
He added: “It was obvious that the situation was extremely serious, the Prime Minister had been gravely ill, colleagues were dealing with huge problems and many were ill or isolating.
“I felt that I should be able to return to work if possible given I was now recovering in order to relieve the intense strain at Number 10.”
Durham Police made a further statement on Monday, to add to the original on Saturday, confirming they spoke to Mr Cummings' father but clarifying that no advice on coronavirus was offered during the conversation.
"Following significant public interest over the last few days, Durham Constabulary wish to add the following to our statement of Saturday, May 23rd," the statement read.
"We can confirm that on April 1, an officer from Durham Constabulary spoke to the father of Dominic Cummings. Mr Cummings confirmed that his son, his son’s wife and child were present at the property. He told the officer that his son and son’s wife were displaying symptoms of coronavirus and were self-isolating in part of the property.
"We can further confirm that our officer gave no specific advice on coronavirus to any members of the family and that Durham Constabulary deemed that no further action was required in that regard.
"Our officer did, however, provide the family with advice on security issues."
There was plenty of reaction to Mr Cummings' press conference from all sides.
SNP Westminster leader Ian Blackford MP said Mr Johnson had “no option” but to sack Mr Cummings, and his failure to do “is a gross failure of leadership”.
Responding to the extraordinary press conference in the Downing Street garden, Mr Blackford said: “What should have been a resignation statement turned out to be a botched PR exercise that changes nothing. It is now beyond doubt Dominic Cummings broke multiple lockdown rules.
“There was no apology and no contrition from Mr Cummings for his behaviour – and now, following this unrepentant press conference, there are no excuses left for him.
“He has done nothing but double down on the double standards he has displayed and which millions of people across the UK are furious about.”
Acting Liberal Democrats leader Sir Ed Davey tweeted: “Cummings’ statement confirms he broke the guidelines. When millions kept to those rules. The PM must now terminate his contract – if he wants to regain any credibility in leading the country on dealing with the Coronavirus crisis.”
A spokesperson for the Labour Party said:“The British people were looking for at least an apology from Dominic Cummings for breaking the lockdown. They got none.
“Millions of people have made extraordinary sacrifices during the lockdown. Families have been forced apart, sometimes in the most tragic of circumstances. They stayed at home to protect the NHS and save lives.
“And yet, the message from this Government is clear: it’s one rule for Boris Johnson’s closest adviser, another for everybody else.”
After Conservative MP Kevin Hollinrake tweeted that Mr Cummings had provided a “detailed and fair explanation of events”, Michael Gove responded: “I agree with Kevin.
Mr Gove, the Chancellor of the Duchy of Lancaster, wrote: “It’s clear now that allegations were made which were untrue and Dominic Cummings acted legally and reasonably.
“Let’s concentrate on the work necessary to deal with the consequences of Covid-19.”
Watch the press conference in full below