  This is an expository-survey on weak stability of bounded linear operators
acting on normed spaces in general and, in particular, on Hilbert spaces. The
paper gives a comprehensive account of the problem of weak operator stability,
containing a few new results and some unanswered questions. It also gives an
updated review of the literature on the weak stability of operators over the
past sixty years, including present-day research trends. It is verified that
the majority of the weak stability literature is concentrated on Hilbert-space
operators. We discuss why this preference occurs and also why the weak
stability of unitary operators is central to the Hilbert-space stability
problem.
