This review presents the state of the art in Gaussian quantum information processing, focusing on continuous-variable systems. Gaussian quantum information processing involves Gaussian states, Gaussian operations, and Gaussian measurements. These are powerful tools for quantum information tasks such as quantum communication, cryptography, computation, teleportation, and state/channel discrimination. Gaussian states are continuous-variable states with Gaussian representations, and Gaussian operations transform these states into other Gaussian states. Gaussian measurements are those that can be described by Gaussian functions. The review discusses the theoretical foundations and experimental implementations of Gaussian quantum information processing, including the phase-space representation, symplectic analysis, and the properties of Gaussian states and operations. It also covers the applications of Gaussian quantum information processing in quantum cryptography, quantum computation, and quantum communication. The review highlights the advantages of Gaussian quantum information processing, including the availability of analytical tools and optical components for Gaussian processes. It also discusses the challenges and future directions of Gaussian quantum information processing. The review is self-contained and provides a comprehensive set of references to the field of Gaussian quantum information processing.This review presents the state of the art in Gaussian quantum information processing, focusing on continuous-variable systems. Gaussian quantum information processing involves Gaussian states, Gaussian operations, and Gaussian measurements. These are powerful tools for quantum information tasks such as quantum communication, cryptography, computation, teleportation, and state/channel discrimination. Gaussian states are continuous-variable states with Gaussian representations, and Gaussian operations transform these states into other Gaussian states. Gaussian measurements are those that can be described by Gaussian functions. The review discusses the theoretical foundations and experimental implementations of Gaussian quantum information processing, including the phase-space representation, symplectic analysis, and the properties of Gaussian states and operations. It also covers the applications of Gaussian quantum information processing in quantum cryptography, quantum computation, and quantum communication. The review highlights the advantages of Gaussian quantum information processing, including the availability of analytical tools and optical components for Gaussian processes. It also discusses the challenges and future directions of Gaussian quantum information processing. The review is self-contained and provides a comprehensive set of references to the field of Gaussian quantum information processing.