The nature of the area of mathematical competitions as a design science is considered, historical roots of mathematical problem-solving competitions are traced, the complementary aspects of mathematics as theory building and as problem solving are touched upon in relation to the practice of competitions. Two historical figures, Euler and Erdos, emerge, and the appropriateness of many of Euler's mathematical ventures are seen as role models for competition mathematics as first put into practice in mathematical competitions towards the end of the nineteenth century. Distinctions and definitions are made, a venture into identifying competition syllabi and the principal types of reasoning employed in solving competition problems is explored, and a description of the many different types of competitions is considered. Interaction between the field of mathematics itself and problem-solving competitions is briefly explored, as are the possibilities that open when competitions and their access to huge amounts of data, both national and international, are taken into account in research belonging to mathematics education. Finally, the range of topics addressed in this special issue of ZDM is covered, along with some possible conclusions relating to the components of the overview.