Number theory is a complex branch of mathematics that conceals many unexplored and unproven conjectures to which mathematicians are trying to find answers and proofs for hundreds of years. In this thesis, we give some of many unproven claims related to perfect and prime numbers. Although they are simple and easy to understand, and therefore interesting to mathematicians and amateurs, they are very difficult to prove. Some of the most famous open questions in this thesis are related to the infinity of perfect numbers, prime and complex Fermat numbers, prime and complex Mersenne numbers and twin primes whose forms, properties and most famous theorems are mentioned. With modern computers, very large numbers of those forms have been discovered, but the strict mathematical proof that there are infinitely many numbers of this form is still missing. The most famous and oldest conjecture is definitely the famous Goldbach‘s conjecture: ”Any even integer greater than 2 can be expressed as the sum of two primes”, which has been a real challenge for mathematicians since 1742. There are many results that point to the exactness of known open questions and many mathematicians, trying to prove these claims, came to other important results and proves, using the results of number theory but also other branches of mathematics. Since they have not been proved for hundreds of years, in this thesis we are not going to answer the famous questions, but we will give some interesting historical facts, show some attempts of proving them and state famous theorems and results given out of the open questions.