This paper presents a comprehensive study of Generalized G-inflation, a framework for the most general single-field inflation models with second-order field equations. The model extends G-inflation and includes previous examples such as k-inflation, extended inflation, and new Higgs inflation as special cases. The authors investigate the background and perturbation evolution, calculating the most general quadratic actions for tensor and scalar cosmological perturbations to determine the stability criteria and power spectra of primordial fluctuations. They also show that the Horndeski theory and generalized Galileons are equivalent, and that non-minimal coupling to the Gauss-Bonnet term can be included in generalized Galileons. The paper begins with an introduction to scalar fields in cosmology, their role in inflation, and the importance of modified gravity theories. It then discusses the properties of Galileons and their covariant extension, which allows for second-order field equations. The authors define the generalized higher-order Galileons and their kinetic gravity braiding, and derive the background equations for the model. They analyze two extreme cases of inflation: one driven purely kinetically and another driven by a scalar potential. The paper then computes the quadratic actions for tensor and scalar perturbations, determining the sound speeds and stability conditions. It evaluates the primordial power spectra for tensor and scalar perturbations, showing that the tensor spectral tilt can be blue if certain conditions are met. The authors also discuss the implications of these results for inflationary models and the potential for non-Gaussianities in the early universe. The paper concludes with a summary of the results, emphasizing the importance of Generalized G-inflation as a generalization of previous inflation models and its potential to provide new insights into the early universe. The authors also note that the model can be extended to non-linear order and that it may have implications for the study of cosmological perturbations beyond linear order.This paper presents a comprehensive study of Generalized G-inflation, a framework for the most general single-field inflation models with second-order field equations. The model extends G-inflation and includes previous examples such as k-inflation, extended inflation, and new Higgs inflation as special cases. The authors investigate the background and perturbation evolution, calculating the most general quadratic actions for tensor and scalar cosmological perturbations to determine the stability criteria and power spectra of primordial fluctuations. They also show that the Horndeski theory and generalized Galileons are equivalent, and that non-minimal coupling to the Gauss-Bonnet term can be included in generalized Galileons. The paper begins with an introduction to scalar fields in cosmology, their role in inflation, and the importance of modified gravity theories. It then discusses the properties of Galileons and their covariant extension, which allows for second-order field equations. The authors define the generalized higher-order Galileons and their kinetic gravity braiding, and derive the background equations for the model. They analyze two extreme cases of inflation: one driven purely kinetically and another driven by a scalar potential. The paper then computes the quadratic actions for tensor and scalar perturbations, determining the sound speeds and stability conditions. It evaluates the primordial power spectra for tensor and scalar perturbations, showing that the tensor spectral tilt can be blue if certain conditions are met. The authors also discuss the implications of these results for inflationary models and the potential for non-Gaussianities in the early universe. The paper concludes with a summary of the results, emphasizing the importance of Generalized G-inflation as a generalization of previous inflation models and its potential to provide new insights into the early universe. The authors also note that the model can be extended to non-linear order and that it may have implications for the study of cosmological perturbations beyond linear order.