The title of the study may shock. Mathematics is the most ancient of the sciences. Why should it be in the service of others, or worse still, in the service of technical activities? In reducing mathematics to a service rôle, does one not belittle its contents, its image? Let us immediately state that in our view ‘mathematics as a service subject’ does not imply some inferior form of mathematics or mathematics limited to particular fields. We mean mathematics in its entirety, as a living science, able – as history has ceaselessly shown – to be utilised in, and to stimulate unforeseen applications in very varied domains. The teaching of mathematics to students of other disciplines must now be accepted as a fact, a social need and, also, a relatively new problematic issue. In this introduction we shall try to show the extent of the phenomenon, the social needs which it expresses, foreseeable developments and likely results in terms of choice of subject matter and teaching methods, and finally the size of the new problems which confront students and teachers alike.