A pipe-cleaning device (1) comprising a cutting head (2) for cutting away material on the interior of a pipe, a body (111) on which the cutting head is rotatably mounted and drive means (3, 4) for driving the rotation of the cutting head (2) with respect to the body (111). In a first aspect the cutting head (2) is provided with one or more inwardly collapsible cutting elements (21) and the body (111) has a plurality of arrays of inwardly collapsible support elements (30) extending in longitudinal direction of the body (111) for supporting the body (111) on the interior of a pipe. In a second aspect the cutting head (2) comprises at least one first collapsible cutting element (21) which extends radially from a rigid central part (11), ends in a first cutting block (22) and is provided with at least one mass increasing element (22) at an intermediate location. In a third aspect, the drive means comprise a stator (3) and a rotor (4), the stator having at least one nozzle (7) for directing a liquid stream successively onto a plurality of drive elements (5) of the rotor (4), the at least one nozzle (7) extending substantially within a plane (V) perpendicular to the rotation axis (9) of the rotor and being located inwardly from the rotor (4).