One of the best types of smart materials is Shape Memory Alloys (SMAs), which have more applications in modern technology such as aerospace, automotive, biomedical, civil engineering, robotics, and so on. Among all parts of them NiTi-based, and Cu–based are more focused on because they have some interesting properties that have made them more widely used in modern technology. In this work, the classifications and applications of SMAs according to the different application fields like biomedical, automotive, and aerospace have been reviewed. Besides the characteristic of (NiTi-based, Cu – based) and the comparison between them in many areas such as: In terms of cost, creation, corrosion resistivity, density, and electrical and thermal conductivity were represented. Each of NiTi and Cu-based SMAs has many useful applications and strengths, but they have some limitations.