A nozzle assembly (1) generates a jet (9) of water, or abrasive particles suspended in water, for use in an abrasive waterjet-cutting head. The nozzle assembly (1) comprises a nozzle element (5, 80, 84, 86, 91) with a tapering bore (7) therethrough, mounted to a carrier (2, 52) so that the bore (7) is connected coaxially to a passage (4) through the carrier (2, 52), the bore (7) and passage (4) preferably having the same diameter where they meet. The nozzle element comprises a superhard material, such as diamond, in the form of a solid body (5, 81, 83, 86) or a coating (90). The nozzle element is mounted to the carrier (2, 52) by a brazed or soldered joint (6), extending normally to a longitudinal axis of the passage (4) and bore (7).