There has been a recent increase in interest in task design as a focus for research and development in mathematics education. This is well illustrated by the success of theoretically based long term design research projects in which design and research over time have combined to develop materials and approaches that have appealed to teachers. One area of investigation is how published tasks are appropriated by teachers for complex purposes and influences mathematics teaching. Tasks generate activity which affords opportunity to encounter mathematical concepts and also to use and develop mathematical thinking and modes of enquiry. Tasks also arise spontaneously in educational contexts, with teachers or learners raising questions or providing prompts for action by drawing on a repertoire of past experience. We are interested in how these are underpinned with implicit design principles. It is important to address also the question of sequences of tasks and the ways in which they link aspects of conceptual knowledge. The communities involved in task design are naturally diverse: designers, professional mathematicians, teacher educators, teachers, researchers, learners, authors, publishers and manufacturers, and individuals acting in several of these roles. We wish to illuminate the diverse communities and methods that lead to the development and use of tasks.