Noncommutative field theory is the analog of classical field theory on quantum spaces. We discuss both classical as well as quantum aspects of such field theories. In particular, a transparent understanding of the crucial phenomenon of UV/IR mixing is obtained using the novel tool of string modes. These exhibit the stringy nature of noncommutative field theory, and will play an important role in the following. In particular, noncommutative gauge theory is defined via Yang–Mills matrix models.