  We develop a general, functional calculus approach to approximation of
$C_0$-semigroups on Banach spaces by bounded completely monotone functions of
their generators. The approach comprises most of well-known approximation
formulas, yields optimal convergence rates, and sometimes even leads to sharp
constants. In an important particular case when semigroups are holomorphic, we
are able to significantly improve our results for general semigroups. Moreover,
we present several second order approximation formulas with rates, which in
such a general form appear in the literature for the first time.
