In this chapter, the authors shows that the class R(X) of Riesz operators on a Banach space X can be characterized in various ways which are simpler than the defining properties. After the rather extensive preliminaries, the authors begin the proof of the asserted characterization of R(X). It should be noted, however, that the properties of Fredholm operators have attracted much interest and they have been the object of extensive study. The authors show that an operator is a Riesz operator if and only if each non zero point of its spectrum is a pole of finite multiplicity. In order to do this, they prove a comprehensive lemma which reveals the role of ascent and descent in a new light. Before turning to West'S proof, it is interesting to note that the decomposition problem has a simple interpretation in terms of the Calkin algebra.