A pontifical university is a university established by the Pope or Holy See.
The University is directly under the authority of the Pope. It is also chartered by the Pope.
A pontifical university grants degrees on the European model (bachelor's, licentiate, doctorate) in theology, canon law, philosophy, and (in the case of Santa Croce) Church communications.
The Pontifical University of the Holy Cross was chartered by John Paul II in 1984 as a pontifical athenaeum. It was elevated to a pontifical university by a grant from Pope John Paul II in 1998.