Both the Roman Catholic Church and the Eastern Orthodox Church claim that the phrase is explicitly referring to a specific church body and/or hierarchy formed by Christ and his disciples of which they, via the Apostolic Succession, are the modern embodiment. Implicit in this belief is the conviction that branches of Christianity which do not claim descent from the apostolic succession are not valid Christian churches but sects.
The tradition within the Protestant churches is more complicated. Anglicans of the 'high church' or 'Anglo-Catholic' tradition define themselves as part of the 'Catholic Communion' associated with the apostolic succession, but not in full communion with Roman Catholicism. Hence like Roman Catholicism and the various Orthodox Churches they share similar beliefs in the sacraments and a use of ritual in church ceremonial. However while many Protestants and the majority of Anglicans also affirm the Nicene Creed, they believe that no one denomination embodies the One Holy Catholic and Apostolic Church referred to in the Creed. Instead they believe that this Church is the aggregation of all true Christians, likely encompassing multiple denominations but whose full membership is known only to God.