  This is an exposition on supercyclicity and weak supercyclicity, especially
designed to advance further developments in weakly supercyclicity, which is a
recent research field showing significant momentum during the past two decades.
For operators on a normed space, the present paper explores the relationship
among supercyclicity and three versions of weak supercyclicity, namely, weak
l-sequential, weak sequential, and weak supercyclicities, and their connection
with strong stability, weak stability, and weak quasistability. A survey of the
literature on weak supercyclicity is followed by an analysis of the interplay
between supercyclicity and strong stability, as well as between weak
supercyclicity and weak stability. A description of the spectrum of weakly
l-sequentially supercyclic operators is also given.
