An improved tubular running tool (14) is disclosed for use on a rotary or top drive drilling rig of the type for inserting and selectively, internally gripping a tubular which may be utilized to lift, lower, rotate, and torque tubulars, and which may be used to fill and/or circulate fluid in and through tubulars within a wellbore. The internal tubular running tool may be used as or in conjuction with fill-up and circulating tools and with cementing head wiper plug assemblies among other tools. The tubular running tool includes an improved moving mechanism (100) having a cylindrical pneumatic chamber (106) annularly positioned with respect to a barrel element (54) that forms an axial fluid pathway therethrough. A cylindrical piston (104) is moveable within the cylindrical chamber to thereby move a cylindrical piston rod (102) connected to gripping slips (62) such that the slips selectively engage an interior portion of the tubular member.