The continuum problem, as formulated by Cantor at the end of the nineteenth century, asks whether there exist infinite sets of real numbers which are equipotent neither to the set N of natural numbers nor to the whole set R of real numbers. This paper summarizes the basic steps of the history of this problem, since its beginning until nowadays, and shows the crucial role it played -and still plays- in the development of set theory. First, the studies that led Cantor to formulate the problem and to conjecture a solution (the continuum hypothesis CH) are presented. The attempts of Cantor and others to prove the hypothesis are also described. Then we mention the works of Gödel and Cohen showing the undecidability of CH with to respect the usual Zermelo-Fraenkelâ€™s axiomatization (ZF) of set theory (and even with respect to ZFC = ZF + choice axiom). Finally we describe some stronger and more recent axiomatizations in set theory, supporting some evidence against CH.