In this thesis a step by step proof of the famous prime number theorem is given. This theorem describes the asymptotic distribution of the prime numbers. The Riemann-zeta function plays an important role in the proof. In the second part of the thesis, the distribution of the zeros of this function is investigated, for which random matrix theory is introduced. The Riemann hypothesis, which hasn't ever been proved, is discussed as well. The famous Clay Mathematics Institute of Cambridge (CMI) has chosen a proof of this hypothesis to be one of the famous seven Millennium Prize Problems.