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 mainstream flow of the working gas passes through the passages formed between adjacent turbine blades. 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 has swirl directions which are co-directionally aligned with the swirl directions of the mainstream flow at the segment.