The phrase universal church is not found in the Bible, although the concept is. The universal church (or simply Church, as a proper noun) refers to all who follow Jesus (Matthew 16:18). As such, there is only one Church.
In some circles, the universal church is also called the catholic church, which is not to imply the Roman Catholic Church. However, it should be noted that the Roman Catholic Church is, by definition, part of the universal church.
A lifelong student of the Bible, Peter DeHaan, PhD, wrote the 1,000-page website ABibleADay.com to encourage people to explore the Bible. His main blog and many books urge Christians to push past the status quo and reconsider how they practice their faith in every area of their lives.
