We prove that there are arbitrarily long arithmetic progressions of primes. There are three major ingredients. The ﬁrst is Szemer´di’s theorem, which ase serts that any subset of the integers of positive density contains progressions of arbitrary length. The second, which is the main new ingredient of this paper, is a certain transference principle. This allows us to deduce from Szemer´di’s e theorem that any subset of a suﬃciently pseudorandom set (or measure) of positive relative density contains progressions of arbitrary length. .