This is a survey paper based on the author’s lectures at the 4th advances course in operator theory and complex analysis. The author reviews the \(H^\infty\)-functional calculus of sectorial operators and related classes of unbounded operators. Their theory is related to the well established theory of \(C_0\)-semigroups and cosine functions. In most cases, the existence of a bounded \(H^\infty\)-calculus is equivalent to certain quadratic estimate arising from harmonic analysis. In addition to describing the topic in general, the paper includes recent results in perturbation theory for functional calculus and for differential semigroups.