  In this paper we survey results concerning the asymptotic properties of
C_0-semigroups on Banach spaces with respect to the weak operator topology. The
property "no eigenvalues of the generator on the imaginary axis" is equivalent
to weak stability for most time values; a phenomenon called "almost weak
stability". Further, sufficient conditions actually implying weak stability are
also given. By several examples we explain weak and almost weak stability and
illustrate the fundamental difference between them. Many historical and
bibliographical remarks position the material in the literature. We conclude
the paper with some open questions and comments.
