\(H^\infty \) -functional calculus for sectorial operators is a topic of paramount importance in operator theory and the harmonic analysis of semigroups, with significant applications in evolution equations and abstract partial differential equations. This chapter presents the related but distinct theory of \(H^\infty \) -functional calculus for Ritt operators. We also introduce the unconditional Ritt property, a natural and important strengthening of the Ritt property, and we explore the profound connections between this notion, \(H^\infty \) -functional calculus and the R-Ritt property.