This paper, presented at the Rencontres Physiciens-Mathématiciens de Strasbourg (RCP25) in 1975, discusses the renormalization of gauge theories, focusing on the Slavnov identities and their implications. The authors, C. Becchi, A. Rouet, and R. Stora, explore the algebraic structure of classical gauge theories and how it is deformed by quantum corrections. They use the Bogoliubov-Paraziuk-Hepp-Zimmermann renormalization theory and the Lowenstein-Lam renormalized quantum action principle to analyze the perturbation series of gauge theories. The paper is divided into two main sections. The first section covers the algebraic discussion of the Slavnov identities, which express the invariance of the Faddeev-Popov Lagrangian under a set of nonlinear field transformations involving Faddeev-Popov ghost fields. The second section discusses a specific model, the SU2 Higgs-Kibble model, to illustrate the operator theory and the physical interpretation of the renormalized theory. Key points include: - The Slavnov identities are proved to all orders of renormalized perturbation theory within the BPHZ framework, assuming a semi-simple Lie algebra and a linear gauge function. - The SU2 Higgs-Kibble model is analyzed in detail, showing that the asymptotic theory is reasonable and the physical S operator is independent of parameters defining the gauge function. - The authors derive the consistency condition for radiative corrections and show that any $\Delta_{\beta}$ can be put in a specific form, leading to the proof of the Slavnov identity. - The physical interpretation of the renormalized theory is discussed, including the connection between parameters in the Lagrangian and physical parameters, and the specification of a physical subspace within the Fock space. The paper provides a comprehensive analysis of the renormalization of gauge theories, emphasizing the importance of the Slavnov identities and their role in ensuring the consistency and physical interpretability of the theories.This paper, presented at the Rencontres Physiciens-Mathématiciens de Strasbourg (RCP25) in 1975, discusses the renormalization of gauge theories, focusing on the Slavnov identities and their implications. The authors, C. Becchi, A. Rouet, and R. Stora, explore the algebraic structure of classical gauge theories and how it is deformed by quantum corrections. They use the Bogoliubov-Paraziuk-Hepp-Zimmermann renormalization theory and the Lowenstein-Lam renormalized quantum action principle to analyze the perturbation series of gauge theories. The paper is divided into two main sections. The first section covers the algebraic discussion of the Slavnov identities, which express the invariance of the Faddeev-Popov Lagrangian under a set of nonlinear field transformations involving Faddeev-Popov ghost fields. The second section discusses a specific model, the SU2 Higgs-Kibble model, to illustrate the operator theory and the physical interpretation of the renormalized theory. Key points include: - The Slavnov identities are proved to all orders of renormalized perturbation theory within the BPHZ framework, assuming a semi-simple Lie algebra and a linear gauge function. - The SU2 Higgs-Kibble model is analyzed in detail, showing that the asymptotic theory is reasonable and the physical S operator is independent of parameters defining the gauge function. - The authors derive the consistency condition for radiative corrections and show that any $\Delta_{\beta}$ can be put in a specific form, leading to the proof of the Slavnov identity. - The physical interpretation of the renormalized theory is discussed, including the connection between parameters in the Lagrangian and physical parameters, and the specification of a physical subspace within the Fock space. The paper provides a comprehensive analysis of the renormalization of gauge theories, emphasizing the importance of the Slavnov identities and their role in ensuring the consistency and physical interpretability of the theories.