Bioabsorbable drug delivery devices including modular drug delivery devices having shapes and sizes adapted to be inserted within a recess on the surface of an implantable prosthesis are disclosed. The devices may be attached to one another to create custom drug delivery devices having controllable drug release characteristics that depend on the composition of individual modules comprising the device. The modules may be cylinders, disks, tiles or tubes comprised of a bioabsorbable polymer and a therapeutic agent. The therapeutic agent(s) may be homogeneously distributed throughout the polymeric body of the device or contained within a cavity within a module comprising the device, or both. The device(s) may be threaded or attached to a prosthesis by a biodegradable adhesive. The modular devices may also be formed into tapered plugs for insertion into a mating receptacle. In another embodiment, the drug delivery device may be inserted within a mesh bag that may be attached to a soft tissue as, for example, by sutures, for localized controlled dispensation of a therapeutic agent.