R-boundedness, for which we refer to Appendix A, plays a key role in the study of Ritt operators and their functional calculus. In the first part of this chapter, we introduce relevant notions similar to some of the ones introduced in Chap. 2 ; this includes R-Ritt operators, R-sectorial operators and R-bounded analytic semigroups. Then, we establish R-bounded analogs of some of the characterization theorems proved in the latter chapter. In the second part of the present chapter, we provide important classes of R-Ritt operators, either on classical or non-commutative \(L^p\) -spaces. In the third part, we prove the existence of Ritt operators which are not R-Ritt and discuss constructions leading to such examples.