UK citizenship fees and language tests will be waived for the Windrush generation and their families, Home Secretary Amber Rudd has said.
She told MPs she wanted people to have the "formal status" they should have had all along, without having to pay naturalisation fees or pass any tests.
She also vowed speedy compensation for anyone who had "suffered loss".
Labour said the "buck stopped" with Ms Rudd for a crisis which had brought "shame" on the government.
In a statement to Parliament, Ms Rudd said the citizenship offer would apply not just to the families of Caribbean migrants who came to the UK between 1948 and 1973, but anyone from other Commonwealth nations who settled in the UK over the same period.
Ms Rudd apologised again for changes to immigration rules - dubbed the "hostile environment" policy - which she said had had an "unintended and devastating" impact on the Windrush generation.
While the public expected immigration rules to be enforced, she said, it had never been the intention for a crackdown on illegal immigration to affect those who were "British in all but their legal status".
"This should never have happened," she told MPs.
"We need to show a human face to how we work and exercise greater judgement where it is justified."
By the BBC's assistant political editor Norman Smith
After days of damaging headlines and repeated apologies, the home secretary sought to draw a line under this scandal with a series of significant concessions.
However, Amber Rudd sparked anger on the opposition benches after she suggested the scandal was the result of successive changes to immigration rules by governments dating back to the 1980s.
While some of the detail surrounding the policy changes have yet be spelt out, ministers will hope that they have done enough to correct the mistakes that have been made and to assuage widespread public anger at the treatment of the Windrush families.
She said she wanted to give the Windrush generation the formal immigration status they "should have had a long time ago" by encouraging those who were not UK citizens to apply to become so.
All fees, language and citizenship tests connected with the naturalisation process would be waived and anyone who had left the UK but been prevented from coming back would now be helped to do so, without any fees.
The offer will apply to those who already have leave to remain, those who do not have the paperwork usually required and children of the Windrush generation.
"In effect this means anyone from the Windrush generation who now wants to become a British citizen will be able to do so," she said.
Ms Rudd said nine residency cases had been settled by a special Home Office team set up to deal with the crisis, while 84 individuals had had appointments made.
Nick Broderick, who came to Britain from Jamaica as a toddler, has spent £30,000 fighting deportation.
He told the BBC that Ms Rudd's announcement was "positive" and would "help a lot of people". He said he was hoping to work again "and get my life back on track".
He said although he had heard nothing about compensation, he was glad he could now become a citizen.
Glenda Caesar, who came to the UK from Dominica with her parents as a six-month-old baby, recently won her battle for citizenship with the Home Office.
The fees and the documentation had been particularly difficult to provide, she said, and she was "so happy" with this move - but still felt annoyed by the saga.
"It's a lot of hurt that I had to go through... We've lost money, I've lost money, I'm in arrears with certain bills.
"So it was, y'know a financial loss to a lot of us, a lot of us, not only myself.
"They've taken 10 years away from me."
All Home Office records dating back to 2002 would be checked to see if anyone had been wrongly deported, Ms Rudd told MPs, adding that no cases had been identified so far with about 50% of documents verified.
She added: "The state has let these people down, with travel documents denied, exclusions from returning to the UK, benefits cut and and even threats of removal. This has happened for some time.
"I will put this right and where people have suffered loss, they will be compensated.
"None of this can undo the pain already endured, but I hope it demonstrates the government's commitment to put these wrongs right, going forward."
Shadow Home Secretary Diane Abbott said the scandal "should not have been a surprise" to ministers, given the warnings they had received, and told Ms Rudd "ultimately the buck stops" with her.
"She is behaving as if it is a shock to her that officials are implementing regulations in the way she intended them to be implemented," she said.
While she welcomed the promise of compensation, she said there was a lack of detail and the sums should "reflect the damage to family life" suffered by families and not be "token".
Reflecting on her families' personal experience, she said: "This was a generation with unparalleled commitment to this country, unparalleled pride in being British, unparalleled commitment to hard work and contributing to society.
"It is shameful this government has treated this generation in this way."
And Labour's David Lammy, a leading campaigner for justice for the Windrush families, said they were being offered something which was already their right.
He called on ministers to help others who also made their life in the UK after World War Two but who might not get equal treatment.
"They are from countries like Bangladesh, Pakistan, India, Nigeria, Ghana and Uganda," he said.
"Many of these people have temporary leave to remain or indefinite leave to remain. It is unfair. They were born under Empire."
Home Secretary set a cut off date of 1973. Many Caribbean countries still had not been granted independence at that stage: Grenada (1974), Dominica (1978), St. Lucia (1979), St. Vincent (1979), Antigua & Barbuda (1981), St. Kitts & Nevis (1983). What about them? #Windrush— David Lammy (@DavidLammy) April 23, 2018