The Boolean network is a succinct and effective tool for describing dynamic discrete models acting on finite sets. However, with the deepening of research and the need of practical problems, traditional Boolean networks have been unable to satisfy the requirements of modeling. Therefore, multi-valued logical networks and mix-valued logical networks come into being, which are collectively referred to as finite-valued dynamic systems(FVDSs). By virtue of the semi-tensor product(STP) of matrices, FVDSs can be converted into equivalent algebraic forms that are easy to deal with. This paper provides a comprehensive survey on the recent developments of STPs and FVDSs. Various generalizations of STPs and their applications are systematically combed. In addition, the latest achievements of FVDSs are emphatically elaborated, involving the current hot issues, the latest research methods, as well as the novel controller design schemes.