The investigation of novel signal processing tools is one of the hottest research topics in modern signal processing community. Among them, the algebraic and geometric signal processing methods are the most powerful tools for the representation of the classical signal processing method. In this paper, we provide an overview of recent contributions to the algebraic and geometric signal processing. Specifically, the paper focuses on the mathematical structures behind the signal processing by emphasizing the algebraic and geometric structure of signal processing. The two major topics are discussed. First, the classical signal processing concepts are related to the algebraic structures, and the recent results associated with the algebraic signal processing theory are introduced. Second, the recent progress of the geometric signal and information processing representations associated with the geometric structure are discussed. From these discussions, it is concluded that the research on the algebraic and geometric structure of signal processing can help the researchers to understand the signal processing tools deeply, and also help us to find novel signal processing methods in signal processing community. Its practical applications are expected to grow significantly in years to come, given that the algebraic and geometric structure of signal processing offer many advantages over the traditional signal processing.
