A shroud assembly for a gas turbine engine, wherein said engine has a temperature at a hot operating condition substantially greater than at a cold assembly condition thereof, said shroud assembly comprising: a supporting structure having an arcuate, axially-extending first hook with a first radius of curvature at a cold assembly condition; at least one arcuate shroud segment adapted to surround a row of rotating turbine blades, said shroud segment including: an arcuate, axially extending first mounting flange having a second radius of curvature at a cold assembly condition; and an arcuate, axially extending first overhang having a third radius of curvature at a cold assembly condition, said first overhang disposed parallel to and radially inboard of said first mounting flange so that said first mounting flange and said first overhang define a first groove therebetween for receiving said first hook; a first interface disposed between said first overhang and said first hook; a second interface disposed between said first mounting flange and said first hook; wherein a selected one of said second and third radii of curvature is substantially different from both the other one of said second and third radii of curvature, and said first radius of curvature; characterized in that: the curvatures of said first overhang and said first mounting flange are selected such that: a first gap is present at one of said interfaces (142, 144) at said cold assembly condition, said first gap closing to a matched interface at said hot operating condition; and a second gap is present at the other of said interfaces (142, 144) at said hot operating condition, said second gap closing to a matched interface at said cold assembly condition. a supporting structure having an arcuate, axially-extending first hook with a first radius of curvature at a cold assembly condition; at least one arcuate shroud segment adapted to surround a row of rotating turbine blades, said shroud segment including: an arcuate, axially extending first mounting flange having a second radius of curvature at a cold assembly condition; and an arcuate, axially extending first overhang having a third radius of curvature at a cold assembly condition, said first overhang disposed parallel to and radially inboard of said first mounting flange so that said first mounting flange and said first overhang define a first groove therebetween for receiving said first hook; a first interface disposed between said first overhang and said first hook; a second interface disposed between said first mounting flange and said first hook; wherein a selected one of said second and third radii of curvature is substantially different from both the other one of said second and third radii of curvature, and said first radius of curvature; characterized in that: the curvatures of said first overhang and said first mounting flange are selected such that: a first gap is present at one of said interfaces (142, 144) at said cold assembly condition, said first gap closing to a matched interface at said hot operating condition; and a second gap is present at the other of said interfaces (142, 144) at said hot operating condition, said second gap closing to a matched interface at said cold assembly condition. an arcuate, axially extending first mounting flange having a second radius of curvature at a cold assembly condition; and an arcuate, axially extending first overhang having a third radius of curvature at a cold assembly condition, said first overhang disposed parallel to and radially inboard of said first mounting flange so that said first mounting flange and said first overhang define a first groove therebetween for receiving said first hook; a first interface disposed between said first overhang and said first hook; a second interface disposed between said first mounting flange and said first hook; wherein a selected one of said second and third radii of curvature is substantially different from both the other one of said second and third radii of curvature, and said first radius of curvature; characterized in that: the curvatures of said first overhang and said first mounting flange are selected such that: a first gap is present at one of said interfaces (142, 144) at said cold assembly condition, said first gap closing to a matched interface at said hot operating condition; and a second gap is present at the other of said interfaces (142, 144) at said hot operating condition, said second gap closing to a matched interface at said cold assembly condition. a first gap is present at one of said interfaces (142, 144) at said cold assembly condition, said first gap closing to a matched interface at said hot operating condition; and a second gap is present at the other of said interfaces (142, 144) at said hot operating condition, said second gap closing to a matched interface at said cold assembly condition.