A lathe machine includes a tailstock unit having a bottom seat driven by a saddle of a turret unit to move longitudinally relative to a headstock by means of a first coupler unit, and a tailstock mount mounted on the bottom seat and driven by a turret mount to move transversely by means of a second coupler unit such that a selected one of a centering mandrel and a drilling tool holder disposed on the tailstock mount is in line with a spindle axis of the headstock. A positioning unit includes blocking and abutment members respectively disposed on the bottom seat and the tailstock mount and having rightward- and leftward-facing inclined surfaces which matingly abut against each other to permit the centering mandrel to be kept in line with the spindle axis, thereby rendering the centering process easy and precise.