There is a class $\mathcal{B}$ of analytic Besov functions on a half-plane, with a very simple description.   This talk will describe a bounded functional calculus $f \in \mathcal{B} \mapsto f(A)$ where $-A$ is the generator of either a bounded $C_0$-semigroup on Hilbert space or a bounded analytic semigroup on a Banach space.    This calculus captures many known results for such operators in a unified way, and sometimes improves them.   A discrete version of the functional calculus was shown by Peller in 1983.