Nowadays, people seem to make up models with true abandon. One school of Bayesians asks that models be linked to observables. DeFinetti’s basic theorem and its many modern extensions is one version. In a little known paper, deFinetti started to build of theory of “almost exchangeability” and its consequences for statistical practice. In joint work with Sergio Baccallado and Susan Holmes (and separately with David Aldous), we try to bring this theory into the modern era. The work uses tools and models from “algebraic statistics” and needs some new notions of convergence. Some new MCMC algorithms allow real examples to be treated.