There is one Latin Rite (Roman Catholic)
A few Eastern rites, who carry some Orthodoxy with them but nothing conflicting with Church doctrine.
Armenian
Byzantine
Chaldean
Coptic
Ethiopian
Maronite
there is a few more.
In his Apostolic Letter, "The Light of the East," John Paul II states,
I believe that one important way to grow in mutual understanding and unity consists precisely in improving knowledge of one another. The children of the Catholic Church already know the ways indicated by the Holy See for achieving this: To know the liturgy of the Eastern Churches; to deepen their knowledge of the spiritual traditions of the Fathers and Doctors of the Christian East, to follow the example of the Eastern Churches for the inculturation of the Gospel message; to combat tensions between Latins and Orientals and to encourage dialogue between Catholics and the Orthodox; to train in specialized institutions theologians, liturgists, historians and canonists for the Christian East, who in turn can spread knowledge of the Eastern Churches; to offer appropriate teaching on these subjects in seminaries and theological faculties, especially to future priests.