A universal firearm adapter configured to mount to the frame of a firearm. The adapter easily, quickly, accurately, securely, concentrically and repeatably, with or without tools, attaches a suppressor or suppressor adapter to the frame or rail system of a firearm. The adapter includes a rail attachment configured to attach to a plurality of dimensionally inconsistent frames/rails and a bore alignment assembly. Moreover, the adapter is configured to axially align the adapter bore with the barrel bore regardless of the dimensional and axial inconsistencies of the barrel and its bore. The bore alignment assembly preferably includes a suppressor mount axially aligned with the adapter bore thereby ensuring that the suppressor is axially aligned with the bore of the barrel.