(on sui juris churches in general)
Liturgical Family: Byzantine
Primary Liturgical Language: Hungarian
Juridical Status: Metropolitan
Approximate Population: A bit over 300,000.
Brief History: Hungary has historically been the crossroads of East and West, so it is unsurprising that the Byzantine Rite has usually had a place despite the general prevalence of the Latin Rite. There was actually an early presence of the Byzantine Rite in the area we now call Hungary, but it kept getting washed away by in-sweeping invasions, with the Tartar invasion being particularly devastating. In the early modern period, however, something more stable and permanent began to develop as Ruthenians and Slovaks of the Byzantine Rite moved into eastern Hungary as refuge from various Turkish invasions. In the seventeenth century, the Ruthenian Catholic Church had begun to develop, and the conditions in the Habsburg empire made it feasible for them to keep their Rite as they moved in and became Catholic. (This is visible in the name of the Church, 'Greek Catholic' being the name that was given under the Habsburgs to Byzantine Rite Catholics of whatever kind.) In addition, Protestants in the area sometimes preferred to convert to Byzantine Rite rather than Latin Rite Catholicism. The numbers therefore began to grow.
As they grew, however, there also developed an active push for the use of the Hungarian language in the liturgy. Indeed, in many places Byzantine Rite Catholics simply began doing the liturgy in Hungarian without the proper approval for it, and the use of Hungarian expanded, despite considerable resistance on the part of Church authorities, throughout the eighteenth and nineteenth centuries. After repeated petitions over the decades, Pius X in 1912 erected the Eparchy of Hajdúdorog; but he explicitly forbade the use of Hungarian in the liturgy itself, requiring them to change over entirely to Greek within three years. For whatever reason, this prohibition was never put into effect -- probably the locals simply dragged their feet, and soon enough there was a World War going on and it would hardly have been a major priority any more.
World War I changed the national boundaries, and thus led to some reorganization of the eparchies in the region. One result was that almost half the parishes of the Eparchy of Hajdúdorog were cut off from the other half; another was that a fair part of the Eparchy of Prešov and a small part of the Eparchy of Mukacheve were cut off from their eparchies; these latter were reorganized as the Exarchate of Miskolc. Unlike the Eparchy of Hajdúdorog, which technically used a Greek liturgy but in reality had a Hungarian one, the Exarchate of Miskolc primarily consisted of churches whose liturgies were based on Church Slavonic. As they acculturated to the new Hungarian nation and interacted with the Eparchy of Hajdúdorog, however, they also began to Hungarianize their liturgy. The redrawing of national boundaries also put Hungarian Greek Catholics into a financial crisis, as many of their institutions were now in countries to which they had little access. Recovery from this sudden impoverishment was very slow.
After World War II, Hungary was taken over by Communists. The Communist regime in Hungary was fairly harsh for Catholics, but unlike most Communist countries, the Hungarian Communist regime did not outlaw the Greek Catholic churches. The Communists constantly interfered, breaking up monasteries and convents, restricting the means of catechesis, and the like, but the churches were not forced underground the way they usually were under Communist rule. Part of the reason for this seems to have been that in Hungary Moscow could not have its preferred option of assimilating everything into a state-dominated Orthodox Church; the Orthodox in Hungary were too scattered and diverse, and the Greek Catholic was unusually large and organized. In addition, getting Greek Catholic clergy who would be willing to switch to a state-backed Orthodox church was not a small task; the clergy as a whole did not have a strong interest in Orthodoxy, despite their Byzantine culture, and it seems to have been thought that active dissolution of the Greek Catholic Church would simply result in the expansion of Latin Rite Catholicism, which from the Communist point of view was not an improvement. It's also possible that by this point the Soviets were less confident in their usual strategy; pushed underground, the Ukrainian Greek Catholic Church had by this point just become even more of a problem for the Communist regime. In 1980 the Eparchy of Hajdúdorog was expanded to include the whole of Hungary, and thus all Greek Catholics within it.
The fall of Communism in Hungary led to an expansion of opportunities for the Hungarian Greek Catholic Church. Institutions began to develop and proliferate. Religious orders returned. In 2015, Francis raised the Eparchy of Hajdúdorog to an Archeparchy, with its seat at Debrecen, and at the same time raised the Exarchate of Miskolc to an eparchy and added another eparchy, Nyíregyháza.
Notable Monuments: The Monastery of Our Lady of Máriapócs is a highly regarded pilgrimage site; it houses a copy of one of the oldest and most important icons of the Hungarian Greek Catholic Church, the Weeping Madonna. (Despite being a copy of the original, which is now in Vienna, it too was a weeping icon.) There are also a great many pilgrimage routes running through Hungary, with some of the sites visited being Greek Catholic.
Notable Religious Institutes: As with most churches that have been Ruthenian at one time or another, the Basilians are quite important, although since they vanished during Communist rule, their return is relatively recent.
Extent of Official Jurisdiction: The Archeparchy of Hajdúdorog and two suffragan eparchies, all in Hungary.
Online Sources and Resources:
http://byzantinohungarica.hu/
http://www.cnewa.org/