Machine tools and machining method

A control unit (72) that relatively moves a headstock (20) and tailstock (30) and a tool (43) to thereby machine a peripheral surface of a workpiece (W) in a radial direction executes control such that a relative feed speed of the tool (43) in the radial direction in a transitional state (T21) where an amount of warpage of the workpiece (W) in the radial direction at a machining position increases is faster than a relative feed speed of the tool (43) in the radial direction in a steady state (T22) where an amount of warpage of the workpiece (W) in the radial direction at the machining position is constant. By so doing, it is possible to reduce a machining time at the time of the start of machining.