The first part of this notes is devoted to the proof of Roth's theorem on arithmetic progressions in the integers whereas a second part gives some short survey on the analogue of Roth's theorem in some infinite subsets of integers of zero density such as the subset of prime numbers. We tried to give the reader all the details needed in the first part so that a master student can read Roth's theorem proof easily. In the second part, some proofs are only sketched and we rather tried to give an idea of the issues specific to zero density subsets than to explain precisely how all the arguments work.