A bent member having a bent portion with an extremely small bend radius such as at most 1 - 5 times its wall thickness and an apparatus and method for its manufacture are provided. A bent member 21 is manufactured by supporting a rectangular tube 16 by a first support means 11 disposed at a first position A while performing relative feeding of the rectangular tube in its lengthwise direction, locally heating the rectangular tube 16 by a heating means 12 disposed at a second position B, cooling the rectangular tube by a cooling means 13 disposed at a third position C, and at the time of working, applying a shear force to the heated portion of the rectangular tube 16 with a shear force applying means 14 disposed in a downstream region D by moving the device 14 two-dimensionally or three-dimensionally simultaneously in the feed direction of the rectangular tube 16 and in a direction generally parallel to a transverse cross section in the lengthwise direction of the rectangular tube 16 which was heated by the heating means 12.