The prime number theorem is that the number of prime numbers less than X is asymptotic to X/log X, conjectured by Legendre, Gauss, and so on. Riemann introduced new ideas into it. The main idea is that the distribution of prime numbers is intimately connected with the zeros of the Riemann zeta function. Following the Riemann's idea, Hadamard and Poussin independently proved the prime number theorem in 1896. In this talk, we introduce the prime number theorem, the theory of the Riemann zeta function and related topics.