The present invention concerns an apparatus for cutting a tubular part (16) extending down into a sea- or ocean bed. The apparatus includes a carrying body (1) adapted to be led over the tubular part (16) that is to be cut, with a cutting assembly placed in the carrying body (1). The cutting assembly includes a ring adapted to be driven in rotation in the carrying body (1). At least one cutting element (6) with a cutting portion (19), is hinged to the ring. At least one actuator (3) is adapted for leading the cutting element (6) between a retracted position and a cutting position against the tubular part (16) that is to be cut. Furthermore it is described an assembly with a cutting apparatus (A) and a carrying body (1) with a least one jaw (15, 28, 29) for fixing and centring the tubular part (16) in relation to the apparatus (A), and a method for subsea cutting of tubular parts (16).