Abstract: A centerless external honing and/or grinding apparatus wherein abrasive material is flexibly mounted around a longitudinal workpiece-receiving recess for flexibly honing and/or grinding the exterior curved surface of a workpiece while it is fed longitudinally forwardly through the workpiece-receiving recess from an input end thereof to an output end thereof, and adapted to impart a desired type of surface finish, usually what is referred to hereinafter as a plateaued surface finish, to a curved exterior surface of the workpiece, which occurs by reason of the flexible mounting of the abrasive material (usually, abrasive particles) and the self-centering action produced by the centerless honing or grinding construction of three similarly longitudinally oriented, laterally and vertically adjacent roller members arranged cross-sectionally in a generally polygonal (usually triangular) array or configuration and with at least one and, in certain cases two, and in other cases, all three, of the longitudinal roller m