Abstract: A damping apparatus and method uses one or more tension elements fabricated from shape memory alloy to cycle through a superelastic stress-strain hysteresis to provide energy dissipation. The method and apparatus can be used for damping of any structures, such as machine parts, building structures, offshore platforms, and nanostructures that are subject to any type of vibration, such as acoustic, seismic, blast, impact, wave and wind. The tension elements can be wire loops or bands that are prestretched into approximately the middle of the stress-strain hysteresis. Such a damping apparatus can be designed to have any selected stroke or force capacity by adjusting the length, thickness and number of the tension elements. Moreover, any type of complex force-deflection hysteresis can be designed, for example, a triangular flag, a rectangle, a rectangular flag, a bowtie, or a staging characteristic that provides for hardening or softening.