In this paper famous Euclidean theorem which is mentioned in the title is considered. Also, numerous proofs of this theorem and its modifications are presented. These modifications were given by famous mathematicians like Kummer, Stieltjes, Fermat, Sylvester, Legendre, Dirichlet, Bertrand and Euler. Proofs are various and they use number theory, analysis, algebra, combinatorics and topology. Indeed, this paper is a complete overview of all relevant fact connected with Euclidean theorem that the set of all prime numbers is infinite and his proof of this theorem.