The Liberal Democrats say it is now time to "regularise" those who have been here longest, so they can integrate into the legal economy and contribute to the exchequer by paying taxes.
They are calling it an "earned route to citizenship", but they also say they have no idea how many people would qualify.
Nick Clegg: "How can you deport someone when you don't know where they live?"
Their manifesto says: "We will allow people who have been in Britain without the correct papers for 10 years, but speak English, have a clean record and want to live here long-term to earn their citizenship. This route to citizenship will not apply to people arriving after 2010."
Probed further, party officials tell me that those who qualify will then have to serve a probationary period of two years during which they would have to work and pay taxes.
Then they would have to pay a fee or do voluntary service as "penance" to qualify for full citizenship.
Would it work?
Such amnesties have been tried before.
Researchers at the Centre on Migration, Policy and Society at the University of Oxford have looked at previous studies, in particular ones of
a United States amnesty in 1986.
They found that "almost all show that the large-scale amnesty implemented in 1986 has not reduced, and has in fact increased, undocumented migration to the US, since it established new migration flows due to networks and family ties".
A total of 2.7m qualified for the amnesty in 1986. By 2000 there were an estimated 9.3m illegal immigrants living in the United States.
Meanwhile, Spain had six amnesties in 20 years. In that time, the number of illegal immigrants applying under the schemes rose from 44,000 to 700,000 - a 15-fold increase.
However, the Liberal Democrats argue there is no link between the amnesties and the later rise in the number of illegal immigrants entering the two countries.
In fact, leader Nick Clegg told Reality Check it was far better to be "smart about this" and get such immigrants "out of the shadows" and into the hands of the tax man.
He also questioned how illegal immigrants could be deported when the authorities didn't know where they lived.
Some of the party's policies do involve tightening up rules. For example they want to immediately re-introduce exit checks - or embarkation controls - on everyone leaving the country.
But there are other relaxations too, like allowing asylum seekers who are still awaiting a decision on their case to take up employment.
The Liberal Democrats say the other parties simply do not have policies to deal with the large numbers of people living in the UK without proper papers.
They also say honesty is their best policy and they point out that in polling immediately after last week's TV debate, Mr Clegg was scoring higher on immigration than Tory leader David Cameron.
A selection of your comments are below.
Spain and Italy have had amnesty for illegal immigrants in the past. Spain's last amnesty was in 2005. In 2010 they still have a major problem with illegal immigration and 20% unemployment. Karl, Antrim
I'm a bit wary of immigration to be honest. It seems to be the one area where all the parties seem a little lost, the claim to know what they are doing but we have seen from previous years that they really don't. This might work for the short term, and might even help lower crime in some areas because illegal immigrants won't be forced to hide, or earn money illegally, however in the long run I fear it may entice more illegal immigrants to try and enter Britain at a later stage. Sean, Medway Towns
It seems to me that this proposal could work - the comparison with USA is not valid as they have a lengthy and leaky land frontier, while Spain is a very short boat trip from Africa. This would ensure that the many immigrant workers on whom we depend in the NHS and elsewhere would be properly established and able to pay taxes here. R.Walshe, Sevenoaks
I think that it's an awful idea. The government should look after people who are living here legally first and foremost. Also this measure would increase unemployment, which is a huge problem at the moment. On top of this other potential immigrants would be encouraged to try to gain illegal entry into our country. Luke, Coventry
Why should someone who is here illegally be allowed to stay? We take our mandated number of legal immigrants. Stop at that. Kevin, Cambridge
I think an amnesty is a good idea, but only if it part of a general plan to manage migration as well as immigration. We do need to have systems in place so that we can track all people coming in to the UK through the normal channels and people leaving. At the present the free market unregulated approach is doing no-one any favours and leading to local crises in planning and provision and consequently fuelling xenophobic tension and extreme right wing politics. Ted, Rotherham
It would be wrong to simply accept that the increase in illegal immigrant populations in the USA and Spain were a direct result of the amnesties. There are all sorts of other migrationary pressures that would have contributed to those increases. It would be interesting to know illegal immigrant populations in countries where no amnesty was given. Lastly, an amnesty will at least allow us to identify these people, and normalise their status, rather than prolonging the current no-man's-land of a policy. Brian, London
Employers use undocumented workers for a reason- they will work for less, and work harder. If you legalize these workers, the demand for workers who will do the work for less in worse conditions remains. The basic rule of supply & demand means more undocumented workers will arrive to fill the gap in the market. Therefore the Liberal Democrat plan is not feasible. Since supply of undocumented workers is unlimited, the best action for government to take is punitive action against those employing. them. A concentration on the demand side of the equation is the only long term route to take. Arif, Luton
No amnesty. It will give the wrong message and the wring result. It's time to start looking after our borders. Steve, Cheshunt
It's clear that the immigration system is in a complete mess, for which Labour and Conservative must share the blame. I'm quite attracted to this idea by the Lib Dems, I'd rather have someone who is here being able to contribute to taxes rather than just living off the black economy and being vulnerable to exploitation by every gang master that comes along. David, Nottingham
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