  The notion of super weak compactness for subsets of Banach spaces is a
strengthening of the weak compactness that can be described as a local version
of super-reflexivity. A recent result of K. Tu which establishes that the
closed convex hull of a super weakly compact set is super weakly compact has
removed the main obstacle to further development of the theory. In this paper
we provide a variety of results around super weak compactness in order to show
the great scope of this notion. We also give non linear characterizations of
super weak compactness in terms of the (non) embeddability of special trees and
graphs. We conclude with a few relevant examples of super weakly compact sets
in non super-reflexive Banach spaces.
