  Only in the last fifteen years or so has the notion of semi-uniform
stability, which lies between exponential stability and strong stability,
become part of the asymptotic theory of $C_0$-semigroups. It now lies at the
very heart of modern semigroup theory. After briefly reviewing the notions of
exponential and strong stability, we present an overview of some of the best
known (and often optimal) abstract results on semi-uniform stability. We go on
to indicate briefly how these results can be applied to obtain (sometimes
optimal) rates of energy decay for certain damped second-order Cauchy problems.
