A floating reamer whose cutting head is axially secured to the shank but movable transversely to the parallel axes of the shank and the cutting head toward and away from a position of coincidence of the axes. Pairs of slidably engaged respective guide faces on one axial end of the shank and on the head are parallel and substantially perpendicular to the axes respectively. Two cutting blades project from the cutting head in opposite radial directions and are normally fixed to the head.