Santo Stefano degli Ungheresi was the former 8th century (?) Hungarian national church, located just south of St Peter's. The site is now part of the Piazza di Santa Marta. There is an English Wikipedia page here.
8th century monasteries
The church was originally founded as a monastery, probably by Pope Stephen III (sometimes given as II) (752-7), as the Liber Pontificum reads:
Tribus monasteriis, quae a prisco tempore in ecclesia beati Petri apostoli ad idem officium persolvendum deputata fuerant, adiunxit quartum.
("To the three monasteries, which from ancient times were deputed to carry out the [liturgical] office at the church of blessed Peter the apostle, he added a fourth.")
The other three monasteries mentioned, clustered around the west end of the Constantinian Basilica of St Peter, were:
Sanctus Stephanus Maior, now the church of Santo Stefano degli Abissini;
Sanctus Martinus in Vaticano, which is thought to be on the site of the south-western crossing pier of the present basilica (there were two other churches on the Vatican with the same dedication);
Sancti Iohannes et Paulus, thought to be on the site of the north-west spiral staircase leading up into the dome of the basilica.
This monastery was, rather predictably, named Sanctus Stephanus Minor to distinguish it from the other one of the same name. Armellini alleges that it was also called Sanctus Stephanus Catabarba Patricia and that the other one was Catagalla Patricia, but Hülsen writes that the two names both belonged to Major. (Incidentally, cata is Greek. The papal curia back then spoke that language.)
However, it must be admitted that the identification with the fourth monastery is a surmise, which assumes that the pope dedicated it to his own patron who was St Stephen the Protomartyr. Back in the Dark Ages and the early Middle Ages, the basilica was surrounded by a swarm of an amazing number of different churches and chapels, and sorting out which was which and where cannot now be a conclusive exercise.
Foundation of Hungarian hospice
King Stephen I of Hungary, the first anointed Christian king of that kingdom, was crowned in 1000 by Pope Sylvester II , and by tradition was granted the old monastery (by that time, monastic life there had certainly failed).
The king restored and enlarged the complex, founded a college for twelve secular canons and also a hospice for Hungarian pilgrims (remote predecessor of the present-day Casa di Santo Stefano). The first documentary reference to it seems to date from 1058, when Pope Benedict X gave it certain privileges.
King Stephen I was canonized in 1083, and the church was re-dedicated to him under the name Santo Stefano dei Ungheresi. Other names documented are Santo Stefano degli Ungari, della Guglia and de Auglia; the last two refer to the obelisk which used to stand nearby until moved to the centre of Piazza di San Pietro.
The Schola Hungarorum or "Hungarian Institution", as it was called, played an important part in maintaining diplomatic relations between early medieval Hungary and the Holy See. It was also a place of learning for Hungarian clerics and intellectuals living in Rome.
Attached to the college and hospice were ancillary buildings such as granaries, store-yards, and mills, and the whole complex was surrounded by a wall. This meant that it functioned economically as a self-contained mediaeval monastery even if not having the status of one.
The Schola Hungarorum was sustained by the income from large estates in the vicinity of Rome. These estates, traditionally granted to King Stephen by the Pope, remained in the possession of the Kingdom of Hungary for hundreds of years. The last one in Celsano was only lost after World War II.
Failure of Schola
In 1205 Pope Innocent III gave the hospice and church into the care of the Chapter of St Peter's, which means that the Hungarian administration must have broken down badly. It was probably then that the church became parochial, a status that is first mentioned in the sources in 1384. By this year, the college of canons had been replaced by a single parish priest.
The church was restored by Emperor Sigismund of Luxemburg (1387-1437) in the early 15th century (he was also King of Hungary), and the complex was given into the care of the Franciscan Friars Minor. King Matthias Corvinus restored the hospice in 1497.
Paulines and the Collegium
Meanwhile, in 1453 the church and monastery of Santo Stefano Rotondo al Celio were granted to the Order of St Paul the First Hermit, also known as the Paulines. This monastic congregation was founded in Hungary in 1250, and had as its patron an Egyptian hermit of the early 4th century (see St Paul of Thebes). They became popular in eastern Europe in the later Middle Ages, although they are not well known in western Europe. At Rome, they were originally at San Salvatore in Onda before the move. In the early 16th century, they were granted the church and hospice at San Stefano degli Ungheresi as well.
From 1541 to 1649, most of Hungary was part of the Ottoman Empire, with Budapest the greatest Muslim city in Europe after Constantinople. This caused the Pauline Order to decline seriously, and by the mid 16th century the monastery at San Stefano Rotondo was moribund. However, a scheme was mooted then by the Hungarian Jesuits (especially by Stephan Szántó) for a seminary to train priests for Hungary, and they were granted that complex in 1579. By then, there was only one Pauline monk left there. Unfortunately, the new Hungarian College was hopelessly underfunded and lasted only one year before being united with the German College to form the German-Hungarian College (Collegium Germanicum et Hungaricum). Hence, the new Collegium inherited San Stefano degli Ungheresi by default.
The Pauline Order revived especially after the Hapsburg Empire conquered Hungary from the Turks, and established themselves at the church of San Paolo Primo Eremita. They are now at Santi Urbano e Lorenzo a Prima Porta.
In the 16th century, St Peter's was rebuilt in the Renaissance style and in the process was greatly enlarged. As a result, most of the buildings of the complex were pulled down because they were too close to the south transept. The church was initially left alone.
However, in 1776 Pope Pius VI embarked on a scheme for an enormous new sacristy for St. Peter's, and expropriated the church which was then demolished. The Pope gave 7500 scudi to the Collegium as compensation for the loss, but predictably it spent the money on other things instead of building a new church elsewhere in Rome. This caused offence, which lasted well into the 20th century.
Hence the Hungarians lost their national church in Rome, but unofficially Santo Stefano Rotondo on the Caelian Hill took over this role, where Pope Pius VI built a new chapel dedicated to King St Stephen of Hungary. Many people think that the whole of that church is dedicated to him, but it is not so.
The church was oriented north to south, and its west side wall corresponded almost exactly to the present west wall of the Palace of the Canonicate which is the tall block to the south of the sacristy. The south-western corner was about half-way down. The north-western corner of the façade was sited just within the westernmost, projecting part of the sacristy itself.
The church was a small basilica, with a nave and aisles but no transept. There was an external semi-circular apse, with a triumphal arch, and the one entrance. The arcades had five columns on each side, and eight of the total of ten were ancient granite spolia. Seven of these ended up in the new sacristy after demolition, while the eighth apparently went astray.
The main altar had an altarpiece showing St Stephen of Hungary, commissioned for the church by the Collegium in 1600.