The limited availability of fossil fuels on the Earth has led researchers to develop new materials that are derived from renewable feedstocks. The polymers produced from the ROP of cyclic esters like (LA and ɛ-CL) are biodegradable, biocompatible, and bioassimilable and thus find major applications in various field. The ROP are catalyzed by the metal-based organometallic catalyst and metal-free organocatalyst. This review exemplifies the living and immortal ROP. The advantage of such polymerization is that they produce polymers with controlled molecular weight distribution. For the immortal ROP, more than one polymer chain grows from the single catalytic site in the presence of chain transfer agents (CTAs), and thus catalyst loading is low, which make the process economically more viable. The nature of CTAs and loading of CTAs with respect to the catalyst is crucial as the catalyst should be effective in the presence of CTAs. The review also discusses functionalized CTAs employed for the polymerization in some instances where functionalized polymers are generated.