This chapter is essentially a treatise on the subject. We shall give all basic material on today’s theory of the distribution of prime numbers; note that we restrict ourselves to asymptotic study, as it is more basic. Thus all deep prime number theorem (Vinogradov’s, Huxley’s, Bombieri’s, Linnik’s) will be presented with complete proofs; analytic means as well are developed with full details. The sieve mechanism stemmed from Linnik and Selberg will be seen as the main mover of our discussion. In the final section, we shall deal with a recent astounding discovery on bounded gaps between prime numbers; our presentation is introductory but will suffice for readers to proceed to the full details.