A rigid pipe-in-pipe structure for subsea transportation of fluids includes inner and outer pipes defining a thermally-isolating annulus between them. Thermal insulation material is disposed in the annulus. The outer pipe is of metal, preferably carbon steel. The inner pipe is a polymeric composite structure of bonded layers including a first, radially outer tubular polymeric electrically insulating layer, which can be of pure polymer, surrounding a second, composite layer including carbon fibers, preferably continuous fibers, embedded in a polymer matrix. Conveniently the second layer is a heating layer in which the carbon fibers conduct electricity to heat the inner pipe. The inner pipe can also comprise a radially inner tubular polymer layer within the second composite layer.