And for nearly two centuries, it has been led by Biondi's Roman Catholic order, the Jesuits.
Soon, it may be led by neither.
Biondi recently announced he intends to retire, and university officials are saying little about the specifics surrounding his departure.
But one thing its board of 50-plus trustees will have to consider is whether to replace Biondi with another Jesuit priest.
That is, if they can find one to take the job.
The "biggest challenge" for a Jesuit institution selecting a new president is that the pool of Jesuits with the right resume is rapidly shrinking, said Fr. Thomas Gaunt, executive director of the Center for Applied Research in the Apostolate at Jesuit-run Georgetown University.
SLU does not make its bylaws public.
But the most recent version -- obtained and published online by the university's faculty senate -- makes it clear that the next president can come from outside the order that founded the school.
The bylaws were amended either in 2006 or 2010 to eliminate the first sentence in Article III, Section 3: "The President shall be a member of the Society of Jesus."
Striking that requirement likely has a lot to do with simple math. The 1960s saw the peak of Jesuit membership in the U.S., with about 7,000 priests. By 1982, that number had diminished to 5,500. Today there are about 2,500 American Jesuits.
In 2001, trustees at Georgetown -- the country's oldest Roman Catholic university -- selected John DeGioia, an alumnus, as its first lay president. He became the first layperson to lead any of the country's 28 Jesuit colleges and universities.
As of July 1, seven of the 28 schools in the Association of Jesuit Colleges and Universities will have lay presidents.
Among the 194 U.S. Catholic colleges that belong to the Association of Catholic Colleges and Universities, 63 percent are led by laypeople. That number is up from about 50 percent in 2001, and 30 percent in 1991.
A month after DeGioia's selection at Georgetown, Marymount University in nearby Arlington, Va., hired a lay president, James Bundschuh, who previously served as dean of SLU's College of Arts and Sciences. The four presidents preceding Bundschuh had been nuns.
Despite those daunting statistics, Jesuit universities are finding men of their own order to run them. Wheeling Jesuit University in West Virginia has had a lay president in recent years but will revert back to a Jesuit leader in July. The school did not conduct a search but promoted a member from its own board.
Regis University in Denver installed a new president last year -- a Jesuit its board recruited from Marquette University, another Jesuit school.
Tom Reynolds, Regis' vice president for mission and ministry, said the board's process was split into two parts. Its initial phase focused on trying to find a Jesuit leader. "If we couldn't find a Jesuit, then we'd open up the search to anyone," Reynolds said.
As SLU now launches its first presidential search in a quarter century, many at the university are hoping for a recruitment process that includes input from faculty, students and administrative staff.
Aside from saying the search process would begin "in the fall," officials have revealed no details publicly about what that search might look like, instead saying in a statement that specifics about the search process "will be communicated to the University community in the weeks ahead."
In the meantime, some faculty members aren't waiting to seek to insert themselves in the process.
Bonnie Wilson, an associate professor of economics, said the SLU faculty would immediately begin meeting to discuss what they'd like in Biondi's successor.
"The community is not sitting back to wait for an invitation from the board," she said. "We presume we'll be invited, and we need to get to work and help ourselves and the board transition for new leadership."