Spectral properties of integral operators on spaces of analytic functions on the unit disk of the complex plane have been studied since 1918. In this dissertation we determine spectral pictures and resolvent estimates for Ces`aro-like operators on the weighted Bergman spaces and show in particular that some of these operators are subdecomposable. Moreover, in a special case, we show that some of these operators are subnormal, some are normaloid, and some are subscalar. We also determine the spectrum and essential spectrum as well as resolvent estimates for a class of integral operators acting on Banach spaces of analytic functions on the unit disk, including the classical Hardy and weighted Bergman spaces, analytic Besov spaces as well as certain Dirichlet spaces and generalized Bloch spaces. Our results unify and extend recent work by Aleman and Persson, [4], Ballamoole, Miller and Miller, [6], and Albrecht and Miller, [3]. In [3], another class of integral operators were investigated in the setting of the analytic Besov spaces and the little Bloch space where the spectra, essential spectra together with one sided analytic resolvents in the Fredholm regions of these operators were obtained along with an explicit strongly decomposable operator extending one of these operator and simultaneously lifting the other. In this disseration, we extend this spectral analysis to nonseparable generalized Bloch spaces using a modification of a construction due to Aleman and Persson, [4].