We present a new approach to the theory of imperfect competition and apply it to study price competition among diﬀerentiated products. The central result provides general conditions under which there exists a pure strategy price equilibrium for any number of ﬁrms producing any set of products. This includes products with multi-dimensional attributes. In addition to the proof of existence, we provide conditions for uniqueness. Our analysis covers location models, the characteristic approach, and probabilistic choice together in a uniﬁed framework.