\indent We will review permutation sequences and allowable permutation sequences and the theorem that every allowable permutation sequence can be realized by an arrangement of pseudolines. \indent We introduce double permutation sequences which provide a combinatorial encoding of arrangements of convex sets in the plane. We shall also review the notion of a topological affine plane and several of its properties. In particular, we show that for every allowable double permutation sequence, there is a corresponding universal topological affine plane P, i.e. any finite arrangement of pseudolines is isomorphic to some arrangement of finitely many lines of P, and that every allowable double permutation sequence can be realized by an arrangement of simply connected sets and pseudoline double tangenets to every pair of these sets. We conclude with some recent results using these methods. \indent All of this is joint work with Jacob E. Goodman and some involves numerous other people including Raghavan Dhandapani, Adreas Holsmen, Shakhar Smorodinsky, Raphael Wenger, and Tudor Zamfirescu.