The particular properties of Shape Memory Alloys associated to their thermoelastic martensitic transformation with hysteresis permits applications of SMA as a damper via the conversion of the work absorbed in each cycle in heat. The guaranteed behavior requires the appropriateness of SMA for the complete requirements of the application. This work shows two complementary aspects of the SMA application in Civil Engineering, the first of them, the SMA dampers in earthquake damping of a family house. The second aspect relates an elementary approach to the damping of stayed cables in bridges, using some data from the Iroise Bridge. The application in the first case needs long time constancy of properties and then around 200 cycles during an earthquake. In the second case, the fatigue life of SMA imposes that only small strains in the alloy can be accepted. Finally, it is emphasized the importance of working conditions, including temperature and time.