There are many Catholics not of the Roman Rite.
Eastern Rite churches are communities of eastern Christians in union with the Holy See. They retain their own distinctive spiritual, liturgical, and canonical traditions. In addition to differences from the Roman rite in liturgy, many of the Eastern Rite churches permit a married clergy. Like the Orthodox Church, they are divided into families; the five major families are the Alexandrian, Antiochene, Armenian, Chaldean, and Byzantine.
- Alexandrian
- Coptic
- Ethiopian
- Antiochene
- Maronite
- Syrian
- Byzantine
- Albanian
- Bulgarian
- Georgian
- Greek
- Hungarian
- Italo - Albanian
- Melchite
- Romanian
- Russian
- Ruthenian
- Slovak
- Ukrainian
- White Russian
- Yugoslavian
- Chaldean
- Chaldean
- Malabâr
- Armenian
The Eastern Rite Churches are Eastern Christian churches consisting of five rites derived from ancient traditions of Christian churches in the East. They are now in communion with the Western church under the papacy. Distinct from both the Orthodox churches and the so-called Independent churches of the East.
After the Great Schism of 1054 between Eastern and Western Christians, some groups, such as the Maronites and Armenians, remained united with the Holy See. Others returned later, by the Brest-Litovsk Union, two Ukrainian Orthodox bishops acknowledged the primacy of the pope. Other groups followed, such as the Chaldeans and other churches of the Byzantine rite (the Ruthenians, the Romanians, and the Melkites). The last were the Malankarese (Antiochene rite) of India. As these various groups of Eastern Catholics grew they established ecclesiastical hierarchies.
The Eastern churches have their own canon law and are not bound by the Code of Canon Law of the Western church. Each church is governed by a patriarch (the patriarchs of Alexandria, Babylon, and Cilicia, and three patriarchs of Antioch). A patriarch with his synod has the highest authority within his jurisdiction and has full authority to appoint bishops and create dioceses.
I hope you find this informative. Currently there are talks with many of the ancient churches concerning reunion with the Holy See.
May the Lord richly bless you.