The essay "50 Years of Horndeski Gravity: Past, Present and Future" by Gregory W. Horndeski and Alessandra Silvestri reflects on the development and impact of Horndeski gravity over the past 50 years. Horndeski gravity, initially formulated in the early 1970s, is a scalar-tensor theory that generalizes Einstein's theory of general relativity by allowing for a non-minimal coupling between the scalar field and spacetime curvature. The theory was rediscovered and widely adopted by cosmologists more than thirty years later, particularly after the discovery of cosmic acceleration in 1998, which prompted the exploration of alternatives to general relativity to explain the accelerated expansion of the universe. The authors trace the historical development of Horndeski gravity, from its initial formulation in 1974 to its modern applications in cosmology. They highlight the complexity of the original Lagrangian and the challenges it posed, but also note that it laid the groundwork for more sophisticated theories. The essay discusses the cosmological implications of Horndeski gravity, including its role in addressing the cosmological constant problem and its potential to describe both the inflationary epoch and the late-time accelerated expansion of the universe. The authors also explore the screening mechanisms in Horndeski gravity, which are crucial for hiding the extra degrees of freedom introduced by the scalar field in dense regions, effectively reducing the theory to general relativity. They discuss the phenomenological consequences of these mechanisms, such as violations of the equivalence principle and modifications to stellar structure and pulsations. Additionally, the essay examines the impact of Horndeski gravity on gravitational wave propagation, including the modification of the speed of gravitational waves and the introduction of additional friction terms. This has implications for the detection of gravitational waves and the study of their propagation on cosmological scales. Finally, the authors highlight the use of effective field theory (EFT) in cosmology, which provides a framework for understanding the dynamics of dark energy and modified gravity in the context of large-scale structure (LSS) surveys. They show how Horndeski gravity can be mapped into an effective field theory action, facilitating the analysis of its cosmological dynamics and the comparison with observational data. Overall, the essay provides a comprehensive overview of the evolution and significance of Horndeski gravity, emphasizing its role in advancing our understanding of the fundamental laws of gravity and the nature of dark energy.The essay "50 Years of Horndeski Gravity: Past, Present and Future" by Gregory W. Horndeski and Alessandra Silvestri reflects on the development and impact of Horndeski gravity over the past 50 years. Horndeski gravity, initially formulated in the early 1970s, is a scalar-tensor theory that generalizes Einstein's theory of general relativity by allowing for a non-minimal coupling between the scalar field and spacetime curvature. The theory was rediscovered and widely adopted by cosmologists more than thirty years later, particularly after the discovery of cosmic acceleration in 1998, which prompted the exploration of alternatives to general relativity to explain the accelerated expansion of the universe. The authors trace the historical development of Horndeski gravity, from its initial formulation in 1974 to its modern applications in cosmology. They highlight the complexity of the original Lagrangian and the challenges it posed, but also note that it laid the groundwork for more sophisticated theories. The essay discusses the cosmological implications of Horndeski gravity, including its role in addressing the cosmological constant problem and its potential to describe both the inflationary epoch and the late-time accelerated expansion of the universe. The authors also explore the screening mechanisms in Horndeski gravity, which are crucial for hiding the extra degrees of freedom introduced by the scalar field in dense regions, effectively reducing the theory to general relativity. They discuss the phenomenological consequences of these mechanisms, such as violations of the equivalence principle and modifications to stellar structure and pulsations. Additionally, the essay examines the impact of Horndeski gravity on gravitational wave propagation, including the modification of the speed of gravitational waves and the introduction of additional friction terms. This has implications for the detection of gravitational waves and the study of their propagation on cosmological scales. Finally, the authors highlight the use of effective field theory (EFT) in cosmology, which provides a framework for understanding the dynamics of dark energy and modified gravity in the context of large-scale structure (LSS) surveys. They show how Horndeski gravity can be mapped into an effective field theory action, facilitating the analysis of its cosmological dynamics and the comparison with observational data. Overall, the essay provides a comprehensive overview of the evolution and significance of Horndeski gravity, emphasizing its role in advancing our understanding of the fundamental laws of gravity and the nature of dark energy.