The record numbers come even as the nation's number of high school graduates begins a slow decline from its peak year of 2009.
Many other area colleges, from Temple University to Pennsylvania State University, also are seeing an uptick.
The boom is fueled by generous financial-aid policies at the elite schools, an improving economy, and an influx of candidates from other countries and California and the Southwest, officials say.
It's also indicative of increasing competitiveness among high school students and a trend of students applying to more colleges - think a dozen or more.
"Getting into Harvard is like winning the lottery twice," said Barmak Nassirian, an executive with the American Association of Collegiate Registrars and Admission Officers. "You're getting into Harvard, and if you don't have the adequate resources, you're getting into an institution that has the financial resources to package you properly."
Harvard requires no contribution from families with annual incomes below $60,000 and no more than 10 percent of income from typical families who earn up to $180,000. Currently, 70 percent receive some aid, the school notes.
Penn's no-loan financial-aid policy, put in place in 2009, is a major factor in the increase, said Eric J. Furda, dean of admissions. The school awards only grants, which don't have to be repaid.
Even higher-income families can qualify.
Of the current freshmen, more than 100 students from families with an income of more than $190,000 received financial aid, with a median award of $16,000, Furda said. Penn's tuition, fees, and room and board top $55,000.
Princeton, which in 2001 was the first to offer no-loan financial aid, showed more modest growth. With 27,115 applications, it was up 3.3 percent, but that follows a 19.5 percent increase last year.
With nearly 34,600 applications - a 32 percent increase - Columbia led the Ivies.
A smaller percentage of students will receive acceptance letters March 30, the day the Ivies announce decisions.
Penn admitted 14 percent of its applicants last year, and this year, that likely will drop to 12 or 13 percent, Furda said. The school will aim for a class of about 2,420, the same as this year, he said.
International applicants grew by 20 percent. More than 4,900 students applied, with Asia accounting for much of the growth, Furda said. The percentage of international students in the United States rose by nearly 3 percent in 2009-10, the Institute of International Education said.
California, where the state system is buckling under budget cuts and demand, continues to be Penn's top state for applications, with 4,169 - a 24 percent increase, Furda said.
Penn also drew 11 percent more from Pennsylvania and 8 percent more from New Jersey.
High school guidance counselors are bracing students for a more difficult year.
"We tell the kids there are no guarantees. You should apply to seven or eight schools - a few that are in reach, a few that are a good fit, and a few that are safety, so you have choices," said Heather Marcus, a counselor at Masterman, a Philadelphia magnet school.
"As far as volume is concerned and demand is concerned, there's no shortage anywhere," Nassirian said.
Neumann College in Delaware County had a 13 percent increase in applications this year. Officials there attribute this in part to more aggressive marketing in South Jersey. Neumann also has more applicants for its new sports management program, in which students can earn a bachelor's and master's in five years, spokesman Steve Bell said.
At Widener, engineering applications rose 32 percent, which school officials attribute largely to a new biomedical engineering degree.
Arcadia, which received nearly 6,000 applications last year, estimates it will be up by about 20 percent. Officials cite Arcadia's extensive study-abroad program, three-year bachelor's degree option, and new master's programs. In 2003, the school accepted three of every four applicants; last fall, it admitted 58 percent.
Penn State president Graham B. Spanier recently told the Board of Trustees that undergraduate applications to all campuses were 6 percent ahead of last year.
"Out-of-state, minority, and international applications are all trending higher, with total international undergraduate applications up 47 percent," Spanier said.
With 3,471 applications, Haverford is up 5 percent, with many applicants from the West Coast.
"Two years ago, California passed Pennsylvania as the second most represented state in our applicant pool, and it is closing the gap with New York for most represented," said Jess Lord, admissions and financial-aid dean.
Some schools, such as Rutgers, say their application pool is about the same as last year. Others reported dips.
The University of Delaware is down 5 percent, said Louis Hirsh, director of admissions.
"The decrease in applications is almost wholly among students with lower GPAs and SAT scores," said Hirsh, whose school received 23,632 as of Jan. 21. "In other words, where our applications are down is among the students who are the least likely to be admissible."
Some schools use "fast-track" applications to increase their draw. Some even visit high schools and offer qualified students admission on the spot.
With its student body at capacity, Ursinus College discontinued fast-track applications and is down this year by about 1,700 - or 29 percent - from last year's record 5,900. It also added a required writing sample.
"After five years, the admission staff has concluded that hundreds, if not thousands, of applicants under this system were not really serious about attending Ursinus," said Richard DiFeliciantonio, vice president for enrollment. "The college boasted application numbers but watched its percentage yield to enrollment on admitted students fall."
Contact staff writer Susan Snyder at 215-854-4693 or firstname.lastname@example.org.