Despite the increasing demands for functional degradable biomaterials, strategies for generating materials with modular compositions and well-defined functionalities from common building blocks are still lacking. Here we report an azido-functionalized cyclic carbonate monomer, AzDXO, that exhibited controlled/"living" ring-opening polymerization kinetics under the catalysis of 1,8-diazabicyclo[5.4.0]-undec-7-ene. Homopolymerization of AzDXO and copolymerization of AzDXO with lactide resulted in polycarbonate and poly(ester-carbonates) with well-defined composition and narrow polydispersity. Further side-chain functionalizations of these polymers were accomplished under facile conditions via copper-catalyzed or copper-free strain-promoted azido-alkyne cyclcoaddition. This versatile monomer building block, obtainable in two steps without tedious purifications, provides a practical solution to the preparation of well-defined functional polycarbonates and poly(ester-carbonates).