We shall discuss two directions of classification of separable Banach spaces up to isomorphism, and their interaction. The first direction is a classification "up to subspaces" as initiated by the work of Gowers: in this direction one looks for a list of "elementary" spaces, such that any Banach space contains a subspace isomorphic to one in the list. The second direction is a classification "by complexity" where each space is characterized by the complexity of the relation of isomorphism between its subspaces.