In this chapter we realize the Cesàro operator as an integral linear transformation on the vector space \(\mathcal {O}(\mathbb {D})\) of all the analytic functions on the open unit disk \(\mathbb {D}\) . This change of view not only helps us see some of the results from previous chapters in another light, but also establishes a rich connection to complex function theory. Furthermore, as we will see in later chapters, this reformulation allows us to prove various spectral properties of the Cesàro operator on an analytic realization of the sequence space \(\ell ^2\) , called the Hardy space. Generalizations of the Cesàro integral operator will be covered in Chap. 11 .