After a review of the classical techniques used in renorming a given Banach space with a locally
uniformly convex norm we will present a unified approach based
on the recent work of M. Raja. Since he has been able to eliminate the probabilistic methods in our
previous results with A.Molto and S.Troyanski he has shwon a new way to construct the norm wich
is valid for dual norms too. The underline idea here will be the topological
notion of network firstly used, in the context of Banach spaces, by R.Hansell for the study of
what he called descriptive spaces, which is a wide class of Banach spaces
for a natural extension of "analytic spaces" in the nonseparable
case. The conection with fragmentability, $\sigma$-fragmentability,
Borel structure and Kadec's renormings will be explained here. The covering
properties related will be analized and the remainig open questions should be presented.
We shall discuss as many applications as possible and we will focus on new lines for research
having in mind the remaining problems. In particular we will explain our transfer
technique which essentially contains the formar constructions involving
a decomposition of the whole space into small pieces in a "descriptive way". The material will
cover, in particular, the research done by our group (A.Molto, V.Montesinos, M.Raja, S.Troyanski,
M. Valdivia and myself) in the last three years.