The Riemann zeta-function ζ(s) is an important object in analytic number theory. The strongest version of the prime number theorem is proved by analyzing the zeta-function as a complex function; in particular, its mysterious zero distribution is an active field of research with plenty of open questions. Also the behavior on the real axis is not yet understood very well. It was a big surprise when Ape´ry proved in 1978 that ζ(3) is irrational. In this chapter we prove Ape´ry’s theorem by his original elementary approach. We also sketch Beukers’ slightly different proof working with multiple integrals.