Fraïssé theory is a method of classical Model Theory of producing canonical limits of certain families of finite structures. For example, the random graph is the Fraïssé limit of the family of finite graphs. It turns out that this method can be dualized, with the dualization producing projective Fraïssé theory, and applied to the study of compact metric spaces. The pseudoarc is a remarkable compact connected space; it is the generic, in a precise sense, compact connected subset of the plane or the Hilbert cube. I will explain the connection between the pseudoarc and projective Fraïssé limits.