The study of dimension groups and their close (and not-so-close) friends leads to a lot of areas. Among them are C*-algebras, Choquet theory, operator theory, number theory (both algebraic and analytic), probability theory (Markov chains and random walks), algebraic geometry (both real and the usual kind), convex polyhedra, topological dynamical systems (classification), measure-theoretic dynamical systems (classification), representations of compact groups, ..., not to mention such outr\'e subjects as logic and semigroups. I will discuss a number of problems, mostly concerning positivity, and how they lead to these connections.