Horndeski gravity, formulated in the early 1970s by Gregory W. Horndeski, has become a cornerstone in cosmological theories. Initially developed as a scalar-tensor theory, it was rediscovered and widely adopted by cosmologists over thirty years later. The theory is characterized by second-order field equations derived from a variational principle involving a scalar field and a Lorentzian metric tensor. Horndeski gravity encompasses various scalar-tensor models, including those with non-linear functions of the Ricci scalar and Gauss-Bonnet terms, and is known for its ability to describe self-accelerating solutions and dark energy dynamics. In cosmology, Horndeski gravity has been instrumental in addressing the cosmological constant problem and exploring alternatives to general relativity. It provides a framework for studying dark energy and modified gravity theories, allowing for the analysis of cosmic acceleration and the dynamics of scalar perturbations. The theory has been tested against observational data, including gravitational wave detections, which have constrained the speed of gravitational waves and provided insights into the viability of different models. Horndeski gravity also incorporates screening mechanisms that ensure consistency with local gravity tests, such as the equivalence principle. These mechanisms allow the theory to avoid violations in dense regions by suppressing the effects of additional degrees of freedom. The theory's ability to produce non-zero shear components in scalar field energy-momentum and differences between lensing and clustering has made it a key signature of modified gravity. The effective field theory of dark energy (EFTofDE) has further solidified Horndeski gravity's role in cosmology, providing a unified framework for analyzing the dynamics of dark energy and modified gravity models. This theory has enabled precise predictions for large-scale structure and gravitational wave propagation, facilitating the exploration of gravity through both cosmological and gravitational wave observations. The combination of Horndeski gravity with the EFTofDE framework has allowed for a comprehensive analysis of the parameter space, ensuring that only viable theories are considered. Overall, Horndeski gravity remains a central tool in the study of gravity and cosmology, offering a robust and flexible framework for understanding the universe's evolution.Horndeski gravity, formulated in the early 1970s by Gregory W. Horndeski, has become a cornerstone in cosmological theories. Initially developed as a scalar-tensor theory, it was rediscovered and widely adopted by cosmologists over thirty years later. The theory is characterized by second-order field equations derived from a variational principle involving a scalar field and a Lorentzian metric tensor. Horndeski gravity encompasses various scalar-tensor models, including those with non-linear functions of the Ricci scalar and Gauss-Bonnet terms, and is known for its ability to describe self-accelerating solutions and dark energy dynamics. In cosmology, Horndeski gravity has been instrumental in addressing the cosmological constant problem and exploring alternatives to general relativity. It provides a framework for studying dark energy and modified gravity theories, allowing for the analysis of cosmic acceleration and the dynamics of scalar perturbations. The theory has been tested against observational data, including gravitational wave detections, which have constrained the speed of gravitational waves and provided insights into the viability of different models. Horndeski gravity also incorporates screening mechanisms that ensure consistency with local gravity tests, such as the equivalence principle. These mechanisms allow the theory to avoid violations in dense regions by suppressing the effects of additional degrees of freedom. The theory's ability to produce non-zero shear components in scalar field energy-momentum and differences between lensing and clustering has made it a key signature of modified gravity. The effective field theory of dark energy (EFTofDE) has further solidified Horndeski gravity's role in cosmology, providing a unified framework for analyzing the dynamics of dark energy and modified gravity models. This theory has enabled precise predictions for large-scale structure and gravitational wave propagation, facilitating the exploration of gravity through both cosmological and gravitational wave observations. The combination of Horndeski gravity with the EFTofDE framework has allowed for a comprehensive analysis of the parameter space, ensuring that only viable theories are considered. Overall, Horndeski gravity remains a central tool in the study of gravity and cosmology, offering a robust and flexible framework for understanding the universe's evolution.