This review paper presents important findings relative to the use of compliant substrates for mismatched heteroepitaxial devices, including the theoretical background, experimental results, and the directions for current efforts. Theories for relative compliance and absolute compliance are presented. Key experimental results are summarized for a number of compliant substrate technologies, including cantilevered membranes, silicon-on-insulator, twist bonding, and glass bonding. Two approaches of current interest, layer transfer and universal compliant trench (UCT) substrates, are presented as potential solutions to the problem of limited absolute compliance in planar compliant substrates attached to handle wafers.