This paper asks two questions about spatial competition. First, does spatial price discrimination lead to lower prices and higher welfare than mill pricing? If firms hold the Bertrand conjecture, the answer is yes for most values of fixed cost. But for the Cournot conjecture, the opposite is true. Second, does the introduction of space into the oligopoly models of Bertrand and Cournot change their results? The answer is generally no; however, spatial Bertrand prices exceed marginal cost, unlike their spaceless counterparts. These answers are obtained using models of spatial competition, of which the Cournot mill pricing models are new.