A shroud segment for a turbine stage of a gas turbine engine forms an endwall for the working gas annulus of the stage. The segment also provides a close clearance to the tips of a row of turbine blades which sweep across the segment. In use a leakage flow of the working gas passes through the clearance gap between the blade tips and the segment. The segment has a plurality of cooling holes and respective air feed passages for the cooling holes. The cooling holes are distributed over that part of the gas-washed surface of the segment which is swept by the blade tips. The cooling holes deliver, in use, cooling air which spreads over the gas-washed surface. The feed passages are configured such that the delivered air opposes the leakage flow of the working gas.