Housing in Rome is expensive and difficult to find. Once you’ve accepted that, you’ll be pleasantly surprised if something should work out.
There are a few housing options set up specifically for lay students at the pontifical universities:
Rumour has it that the Anglican Church of St. Paul Within-the-Walls (on Via Nazionale, near the Angelicum) still rents rooms to students, but that there is a waiting list and/or Anglican students are given preference. There is nothing listed on the St. Paul’s website about this, but you might try contacting the church directly.
The Angelicum has a student housing "office" that publishes a list of apartments available for students. This list can be found at the Ang itself, in the building that houses the Theology Dean’s office, on a bulletin board on ground floor. Students from other schools make use of it also. If you aren't in Rome or can't easily access that list, here is contact info for the student housing office:
Posting a request on the Facebook group for lay students may actually be the best way of finding a living situation. Another international Facebook group, "Expats Living in Rome," has many apartment listings (and job listings). You might also look at Rome Craigslist or Wanted in Rome for classifieds on apartments.
A growing number of lay students have begun renting apartments in Santa Marinella. This small town on the beach is about an hour from Rome by train. Since it is a summer beach town, landlords rent apartments during the school year at low rates, simply to have some income from the apartments. (However, some landlords, seeing that "Nella" is becoming a popular place for foreign students, have begun to raise rents.) As the town is small, it’s easier for lay students to get together socially and have a sense of community. Students with families often find this an ideal situation. So do students at Regina Apostolorum, since the school is just as close to Santa Marinella as it is to Rome (at least in terms of travel time).
Students at the pontifical universities living in Santa Marinella have started a Facebook group, “Americans (and more) in Santa Marinella.” Certain rental agencies in Santa Marinella are more student-friendly than others, so it would be wise to ask the Facebook group which rental agencies students are currently using. There may also be apartments or rooms already available with other students.
Housing Options for Women
There are many convents which offer housing to female students. The living situations and prices can differ greatly from one place to another. Usually meals are included, and there is almost always a curfew. In some of these places, the sisters don’t care if you stay out—you will simply be locked out. In others, the majority of girls are young Italian students, so the sisters take a motherly role and frown on anyone staying somewhere else. Often you will be assigned a roommate, as single rooms are rare. Here is a list of convents that offer hospitality; many cater to tourists but ones that say "pensionato universitario" cater to students. Ask others on the Lay Students Facebook page for more ideas on convent housing.
The Apostles of the Interior Life (Apostole della Vita Interiore) have an apartment off Via Flaminia that they rent out to female students. The number of places available is, of course, limited. Contact them for further information.
Some students find both housing and work as an au pair. This can be a substantial savings, but also an overwhelming time commitment for a student. See the page on “Funding your Education” for more information on this.
Housing Options for Men
The Diocese of Rome used to publish a list of religious institutes offering housing for male students (and a much longer list for female students). These are not currently available, so ask on the Lay Students Facebook page about "convitti" for male students if you are interested. These are usually for priests studying in Rome and may be expensive, but you might find a reasonable deal.
The Casa Balthasar offers a program where “young Christian students of all nationalities, who aspire to give themselves totally to the Lord in one or another form of consecrated life, live or gather daily. While studying at universities in Rome, they wish to live a time of discernment and election. The Casa Balthasar provides an extra-academic spiritual and intellectual formation, in the spirit of the Gospel and the Christian tradition.” Obviously, this is not simply a living arrangement. However, if a young man is coming to study in Rome with the goal of discerning the priesthood or religious life, this community could be an ideal place. Pope Benedict XVI was involved with the founding of this house when he was Cardinal Ratzinger.
Temporary Housing Arrangements / Pensione
Finally, if you are looking for temporary housing for yourself or for friends/family who are visiting, pensione run by religious communities are a good option. The American parish in Rome, Santa Susanna, has a list of pensione and prices. And here's a list from a Jesuit website. Some of the religious houses on these lists may also offer long-term housing for female students.
Of course, you can always try www.hostelworld.com for hostels both within and outside Rome. They are usually cheaper than convents and have no curfews; on the other hand, the atmosphere is a bit different!