Prime Number Theorem: Riemann zeta function, zero free region, proof via a Tauberian theorem and elementary proof with the method of Bombieri-Wirsing. Sieves: Selberg's lambda square method. Brun-Titchmarsh theorem. Brun's result about the twin primes. Romanov Theorem. Sumsets: Schnirelmann's notion of density. Mann's theorem. Existence of an asymptotic basis for sets having positive density. The Schnirelmann's result about the Goldbach problem. New proof of the Romanov's theorem. Set of integers and linear progressions: van der Waerden's theorem. the conjecture of Erdös- Turàn and the Roth's theorem. Highlights about the Szemerédi's proof of the conjecture. Linear progressions of prime numbers: existence of triplets of primes (via Roths's theorem) and highlights about the solution of the general problem by Green e Tao (via the Szemerédi's theorem). The Waring problem: qualitative solution of Linnik and Newmann. Highlights about the quantitative aspects (singular series and the Hardy-Littlewood result).