Mr Cameron said that the provisions in the Open Public Services White Paper, published today, would mark the first step on the road to a "better, fairer country" in which people enjoy more choice, less bureaucracy, improved services and equal access for rich and poor.
But his proposals met a furious response from the unions, who accused the coalition Government of seeking to break up the public services.
Speaking in east London as the White Paper was unveiled to the House of Commons, Mr Cameron said he was "absolutely determined" to see through wholesale reform of the public services.
Under the changes, every public service except for national security, frontline policing and the judiciary will be opened up to providers from the private and voluntary sector.
The White Paper floats proposals to enshrine in law "a general right to choose" in areas like education, health, social care and housing, with new powers for the ombudsman services to act as an "enforcer of choice in public services".
And consumer organisation Which? will extend its role as advocate for consumers in the private sector to cover the public sector too.
"This White Paper says loud and clear that it shouldn't matter if providers are from the state, private, or voluntary sector - as long as they offer a great service," said Mr Cameron.
And he added: "The old narrow, closed, state monopoly is dead."
Unveiling the proposed reforms in the House of Commons, Cabinet Office minister Oliver Letwin said they would provide "increased choice and power for service-users, so that we can provide access to excellence for all".
At present, the wealthy are able to pay for good-quality services where the state sector provision is not up to scratch, while the poor are forced to accept whatever they are given, said Mr Letwin.
And he told MPs: "We need to give everybody the same choice in and the same power over the services they receive that well-off people have already."
The reforms would be driven by the principles of choice, decentralisation, diversity, fair access and accountability, he said.
This would include personal budgets for social care users by 2013; funding following the user in state schools, universities, childcare and the NHS; premium payments for school pupils and healthcare patients from disadvantaged backgrounds; easily accessible information about public service performance; and payment by results for providers of services such as welfare-to-work, offender rehabilitation, and drug and alcohol treatment.
Mr Cameron said the launch of the reforms was an "important day" for the country and the coalition Government, and their impact would be felt "in every state school, hospital and prison, by every doctor, teacher, parent, patient and citizen".
Today's White Paper was initially expected to be published in February, when Mr Cameron provoked union fury by promising a "presumption" that private providers could run all but the most sensitive public services.
The delay fuelled speculation about bitter fights within the Government over the major policy initiative, including reports that Mr Cameron's policy guru Steve Hilton came close to quitting.
But while the language may have been toned down, Mr Cameron insisted he has in no way watered down his plans.
"I know there are those who thought we might be pulling back or losing heart for the task ahead," he said.
"So let me assure you of this: we are as committed to modernising our public services as we have ever been. I'm not going to make the mistakes of my predecessors - blocking reform, wasting opportunities and wasting time.
"This is a job that urgently needs to be done, and we are determined to see it through."
He described public services as "the backbone of the country" but complained that they still operate with a "take-what-you're-given" philosophy that has failed sufficiently to close gaps between the life quality of the rich and poor.
"I know what our public services can do and how they are the backbone of this country. But I know too that the way they have been run for decades - old-fashioned, top-down, take-what-you're-given - is just not working for a lot of people," said the Prime Minister.
"Public services were centralised with all the right intentions: to drive progress through from on high, to keep tabs on how that progress was going with targets and rules and inspections. But the impact of this has been incredibly damaging."
Public services were "failing on fairness", with people in the poorest neighbourhoods dying seven years earlier than those in the richest; children from disadvantaged families half as likely to get five good GCSE passes as their better-off contemporaries; and just 40 students on free school meals getting one of the 80,000 places at Oxford or Cambridge universities, said Mr Cameron.
"And we're not getting value for money either," he said. "Total public spending increased by 57% in real terms from 1997 to 2010. But on no measure can we claim that things have improved by more than 50%.
"Even if we weren't deeply in debt, we would have a responsibility to do something about this."
Setting out his vision for the future of public services, Mr Cameron said: "It's about ending the old big government, top-down way of running public services and bringing in a Big Society approach; releasing the grip of state control and putting power in people's hands.
"The old dogma that said Whitehall knows best - it's gone. There will be more freedom, more choice and more local control."
Mr Cameron insisted that diversity was not a "threat" to essential services, but "a promise of better public services for everyone".
But unions warned that the Government's plans - published on the day that private care home company Southern Cross announced it was closing down - would put quality public services at risk.
TUC general secretary Brendan Barber said: "This is nothing less than a manifesto to break up our public services, smuggled out while all attention is focused on the misdeeds of News International.
"Yet on the day Southern Cross's failure underlines just how dangerous introducing the profit motive into public service can be, people should be very afraid at what these proposals could mean.
"Of course they are skilfully wrapped up in warm words, but when the Prime Minister talks of charities and voluntary groups, he means parcelling up public services for private companies.
"When he talks of ending top-down control, he really means introducing a postcode lottery with few winning tickets, and when he talks of fairness he means new opportunities for the sharp-elbowed middle classes to push others aside."
Unison leader Dave Prentis said: "If this is Cameron's big idea, he needs to go back to the drawing board. Today's White Paper is just another stepping stone towards Cameron's same old goal to privatise public services.
"The collapse of Southern Cross today gives a clear warning as to what happens when you let the market run public services.
"The sale of elderly care homes started out small, but it wasn't long before private equity companies got their teeth into the market. Now that the profits are less, they want out, and thousands of elderly residents and their families are bewildered and angry over their uncertain future.
"Not only does Cameron want to wash his hands of providing public services, he wants cut-price privatisation. He is removing protections on workers' pay and conditions, creating a race to the bottom, so that services are more attractive to private buyers."
Unite assistant general secretary Gail Cartmail said the "utterly wrong-headed" proposals would involve passing "vast parts of the public realm into uncertain hands".
"The public are not fooled. They know that this is not about improving service quality. Service quality was sacrificed the day George Osborne cut council budgets by 28%," said Ms Cartmail.
"This is entirely about shrinking the society we have built up through our taxes and the endeavours of working people in the generations since the war to build a fairer Britain where quality services were available to all."
Mark Serwotka, leader of the Public and Commercial Services union, said: "This has nothing to do with people power, it's about handing more of our public services over to private companies so they can make massive profits at taxpayers' expense."
And Bob Crow, leader of the Rail, Maritime and Transport union, said: "This Government would privatise the air that we breathe if they thought that they could get away with it.
"If David Cameron seriously thinks he can flog off what's left of the family silver to the same greed merchants who created this crisis, then he will have the fight of his life on his hands."
Paul Kenny, leader of the GMB union, said the reforms amounted to "a dangerous re-threading of the bald tyre that led to today's crash at Southern Cross" and "a formula for taxpayers' cash ending up in offshore tax havens".
Mr Kenny said: "That the Government should seek to dress up this extension of privatisation on the day Southern Cross suspended its shares shows the extent to which it has lost touch with the real world."
Sir Merrick Cockell, chairman of the Local Government Association, said: "Public services are more accountable, more efficient and more in tune with the needs and aspirations of local people when they are delivered locally. Now ministers need to deliver on their commitment to let go and devolve services down to the local level.
"We urge Government not to hold back. No department should consider itself above the need to break up the centralised power they have held over local areas.
"It is vital that the devolution of power from Whitehall down to town halls and communities is comprehensive and not tied up in red tape."
Co-operative Party general secretary Michael Stephenson said: "Cameron's faux mutualism is not 'people power' - it's simply bogus.
"The thinking that should have been done clearly has not, so the accountability mechanisms to make public services genuinely democratic are not there.
"Oliver Letwin today proposes privatisation and calls it 'creating employee mutuals', just as 20 years ago he proposed the poll tax and called it a 'community charge'. As usual, the small print does not match up to the Cameron rhetoric."
Shadow Cabinet Office minister Tessa Jowell said: "This White Paper contains few new ideas and even fewer new proposals. Having promised radical change, the Tory-led Government are lagging behind their earlier rhetoric and are yet to catch up with the last Labour government.
"Our public services face significant challenges over the coming years, not least as a result of cuts that are too far and too fast. Yet it appears that the Government are simply obsessed with spin and presentation rather than providing the reform that our public services need."
But Chief Secretary to the Treasury Danny Alexander described the White Paper as "an important milestone in the Government's mission to make public services better for the people that use them".
"This paper alone won't change people's lives, but the principles it sets out will shape and improve the public services people use for generations to come," said Mr Alexander.
"It will give people more choice, make services fairer, devolve power from the centre and make services more accountable to the people that use them."
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