  This article summarises the theory of several bounded functional calculi for
unbounded operators that have recently been discovered. The extend the
Hille--Phillips calculus for (negative) generators $A$ of certain bounded
$C_0$-semigroups, in particular for bounded semigroups on Hilbert spaces and
bounded holomorphic semigroups on Banach spaces. They include functions outside
the Hille-Phillips class, and they generally give sharper bounds for the norms
of the resulting operators $f(A)$. The calculi are mostly based on appropriate
reproducing formulas for the relevant classes of functions, and they rely on
significant and interesting developments of function theory. They are
compatible with standard functional calculi and they admit appropriate
convergence lemmas and spectral mapping theorems. They can also be used to
derive several well-known operator norm-estimates, provide generalisations of
some of them, and extend the general theory of operator semigroups. Our aim is
to help readers to make use of these calculi without having to understand the
details of their construction.
