The axiom of choice is the most controversial axiom of mathematics. Together with the continuum hypothesis it formes the first of Hilbert's problems. Though the logical status of these two statements has been settled by the work of Goedel and Cohen, the question of their truth (or, if you prefer, validity) remains unsettled. This situation leads to several fundamental questions regarding the foundation of mathematics. E.g.: Should the axiom of choice be regarded as valid - as done in almost all modern mathematical textbooks ? If not, should or can it be replaced by some alternative axiom ? Is there only one mathematical truth ? If familiar theorems collapse under the rejection of the axiom of choice, could it also happen that desirable statements that are false in the presence of the axiom of choice become true in its absence ? The purpose of the talk is to present a short history of the problems involved, to outline the relationship between the axiom of choice and the continuum hypothesis and to discuss such questions as those raised above.