It is still an open question to know whether a space which is uniformly or
even Lipschitz homeomorphic to c0 is actually linearly isomorphic to c0. However
recent progress have been achieved on thsi question by N. Kalton, G. Lancien
and myself, following the seminal article of W. B. Johnson, J. Lindenstrauss and
G. Schechtman. An account of this progress will be provided in these lectures.
The link with transfinite arguments, and in particular with the Szlenk index,
will be explained. The somewhat unexpected connection with renormings enjoying
the uniform Kadec-Klee property will be displayed. The non separable theory,
which significantly differs from the separable one, will also be presented.