There will be tough competition for places this summer
An extra 42,367 people have applied to do full-time undergraduate courses across the UK this year: about four times the extra places available.
Latest figures from the admissions service Ucas show an average rise of 8.8% against this time last year.
Many new universities are reporting an "unprecedented" surge - with warnings that this will mean turning many away.
The government said it had funded huge growth in higher education but everyone had to tighten their belts.
But the National Union of Students (NUS) has called for a review of student places.
Applications are up by between 6% and 37% in the Million+ group of 27 new universities in England and Scotland.
Its findings suggest students completing their A-levels, Highers and Advanced Highers this year could face a hard time when looking for places for the autumn - and that many could miss out.
Cap on expansion
The Ucas statistics show that, as of 24 March, 524,151 people had applied to start full-time undergraduate courses across the UK this autumn compared with 481,784 at the same point last year - an extra 42,367 people (up 8.8%).
The service said the pronounced increase in mature applicants continued: up 14.7% among those aged 21 to 24 and 15.8% among the over-25s.
Universities are fined if they over-recruit among EU students. Non-EU students make up 35,074 of the total applicants, almost 10% more than last year.
After years of urging a widening of participation in higher education, the government now faces calls for extra funding to meet the surge in demand.
England's Department for Innovation, Universities and Skills (Dius) is funding an extra 10,000 places in 2009-10 but it had originally estimated having 15,000 more.
It had to make a cut after learning that many more students than anticipated would qualify for financial support.
Grants to universities by the Higher Education Funding Council for Wales are going up by just 1.7% overall - meaning cuts for some institutions.
In Scotland almost a third of institutions face a funding cut in real terms for 2009-10.
Pam Tatlow, head of the Million+ group, said it would be "unthinkable" if thousands of suitably qualified applicants could not enter higher education - with individual institutions reporting increases of 15%, 25% and 37% on this time last year.
And there have been wider worries about funding following yesterday's Budget.
Universities UK and the 1994 Group, representing research-intensive universities, expressed concern about £400m in savings announced by Dius and there are cuts to the devolved settlements for the rest of the UK.
England's Higher Education Minister David Lammy said that far from freezing or capping student numbers, the government had presided over a huge increase, backed by record investment: more than £7.5bn this year.
"Yesterday's Budget changes none of this but it is right that as the nation tightens its belt in the face of real pressures on public spending, the higher education sector plays its part in ensuring value for money for taxpayers by finding efficiency savings.
"We are targeting resources at frontline provision and will not make the mistakes of past recessions which is why we are doing everything we can offer real help to our young people through education, training and financial support."
Call for review
The chief executive of Universities UK, Diana Warwick, said the application figures showed "unprecedented levels of interest" in higher education.
But she added: "We expect a challenging admissions period this summer due to the restriction on the number of undergraduate places that will be funded this year.
"This means that competition for places will be particularly fierce for those courses in high demand."
The NUS president, Wes Streeting, said: "Unless there is an urgent expansion of places, universities will be unable to meet this demand.
"We are therefore calling on the government to launch an immediate review of student numbers for the coming year and invest in the number of places needed to guarantee a place to those who have the ability and aspiration to succeed in higher education."
Shadow universities secretary David Willetts said: "Ministers are paving the way for thousands of young people to be disappointed this summer, forcing them onto an appalling job market.
"Young people must not be the victims of this recession."
Liberal Democrat spokesman Stephen Williams noted that the increased interest was coming not just from school leavers but from older people who were bearing the brunt of the economic crisis.
"If ministers hadn't botched up the funding last year, more of these people would be able to get the skills they need to secure themselves a better future," he said.