Theoretically speaking, black holes are just one of the many roots to one certain quantum physics equation. The mathematicians and scientists involved in that research didn't expect to theorize a black hole; in fact, they were actually trying to solve another problem. When black holes were first introduced to the public, people instantly believed the theory because it came from the top international scientists. Because of the hype and hope people had over these things, astronomers suddenly had an extra responsibility of finding them.
So to answer your question, black holes do not necessarily exist, but there are many strong clues that suggest they do. Believe what you want