The SMA was studied for their macroscopic application in damping for civil engineering. The study is a synthesis and includes an outline of the models required for the SMA simulation and some case studies using the finite element analysis methods. This work is an overview that focuses in the mitigation of the oscillations in structures induced by earthquakes, and for a reduction of the oscillations amplitude in stayed cables under the action of rain, wind or traffic. The analysis needs the required conditions for each application determining the working conditions. The study includes the number of working cycles, the temperature effects and the cooling actions and, for instance, the action of the cycling frequency. The main target relates the appropriateness of the SMA for each purpose, and the suitability of the SMA device is always experimentally guaranteed. Furthermore, the applicability of the obtained results for SMA and the practical behavior of the SMA dampers were studied in international facilities. The paper includes appropriate suggestions for a correct preparation of the SMA dampers. This work outlines the effects of stress and temperature aging in NiTi, describes the particular structural effects between 18R and 6R, introduces a first attempt in the dynamic properties of the CuAlBe single crystals and summarizes some recent suggestions for damping using SMA