SMA and superelastic materials (SEM) are mostly used for low force/large displacements applications involving tensile or bending modes of loading. The paper describes experiments on radial compression of SMA and superelastic wires and tubes between flat loading surfaces. Elastic (recoverable) deformations up to 22.5% with high damping under loads in the KN/mm range for solid wires and 10-100 N/mm range for tubes of OD = 0.4-2 mm have been observed. The recoverable deformation as great as 60% has been measured on a thin-walled SEM tube. This phenomenon was named a "Giant Superelasticity Effect" (GSE). The experiments were performed in a broad range of loading rates. Stainless steel specimens were also tested for comparison. A close-form analysis for thin-wall tubes and some realized and potential applications of GSE are described.