Mathematical tasks and tools, including tasks in the form of digital tools, are key resources in mathematics teaching and in mathematics teacher education. Even so, the "design" of mathematical tasks is perceived in different ways: sometimes seen as something distinct from the teaching and learning process, and sometimes as integral to it. Whilst task design has often been carried out by designers or mathematicians (perhaps as textbook authors), the focus for this review article is research that has involved mathematics teachers as partners in the design of tasks. The article provides a state-of-the-art review of relevant literature and is presented under three headings that consider, in turn, the role of mathematical "tasks"; the nature of "task design"; and the notion of "partnerships for task design" in mathematics education. Subsequently, we present current research that is providing new insights into tasks, task design, and task design partnership. Based on this, we argue that "task design" needs to pay particular attention to "what" to design, "which" tools are necessary or beneficial, and under "what" conditions; digital tools and task resources offer particular affordances that traditional resources cannot provide; and not only do teachers benefit from being partners in task design (in terms of their professional learning) but without their involvement some aspects of task design would most likely be neglected.