A turbine shroud adapted to extend around a bladed turbine wheel to block gasses from passing over the bladed turbine wheel is disclosed. In a segmented embodiment, each turbine shroud segment may include a carrier segment and a blade track segment. The carrier segment may comprise metallic materials and may be formed to define an attachment-receiving space. The blade track segment may comprise ceramic matrix composite materials and may be formed to include an attachment portion that extends radially outward from the runner into the attachment-receiving space formed by the carrier segment.