The geometric theory of Banach spaces underwent a tremendous development in the decade 1990-2000 with the solution of several outstanding conjectures by Gowers, Maurey, Odell and Schlumprecht.

Their discoveries both hinted at a previously unknown richness of the class of separable Banach spaces and also laid the beginnings of a classification program for separable Banach spaces due to Gowers.

However, since the initial steps done by Gowers, little progress was made on the classification program. We shall discuss some recent advances due to V. Ferenczi and myself on this by means of Ramsey theory and dichotomy theorems for the structure of Banach spaces. This simultaneously allows us to answer some related questions of Gowers concerning the quasiorder of subspaces of a Banach space under the relation of isomorphic embeddability.