This lecture by L. D. Faddeev provides a detailed introduction to the Algebraic Bethe Ansatz (ABA) for integrable models, focusing on the XXX model of spin 1/2. The ABA is a method for solving quantum integrable models, which are solvable through the quantum inverse scattering method. The lecture begins with a review of the classical inverse scattering method and its quantum counterpart, emphasizing the importance of integrable models in various fields such as condensed matter physics and string theory. The core of the lecture is the presentation of the XXX model, which is defined on a discrete circle with periodic boundary conditions. The model is described using a Lax operator, which is a key object in the quantum inverse scattering method. The lecture derives the Bethe Ansatz equations (BAE) for the XXX model, which are algebraic equations that determine the eigenstates and eigenvalues of the Hamiltonian. These equations are derived using the commutation relations of the Lax operator and the properties of the monodromy matrix. The lecture also discusses the thermodynamic limit of the XXX model, where the number of sites \( N \) approaches infinity. In this limit, the BAE simplify, and the spectrum of the model can be described in terms of quasiparticles with momentum and energy. The lecture explains how these quasiparticles interact through scattering processes, and how the Bethe Ansatz equations can be used to describe bound states and complex solutions. Finally, the lecture provides explicit expressions for the eigenvalues of important observables, such as the spin and momentum, on Bethe vectors. It also discusses the physical spectrum of the ferromagnetic phase of the model, including the ground state and excited states, and the scattering properties of the quasiparticles. The lecture concludes with a discussion of the mathematical and physical features of the thermodynamic limit, highlighting the differences between even and odd values of \( N \).This lecture by L. D. Faddeev provides a detailed introduction to the Algebraic Bethe Ansatz (ABA) for integrable models, focusing on the XXX model of spin 1/2. The ABA is a method for solving quantum integrable models, which are solvable through the quantum inverse scattering method. The lecture begins with a review of the classical inverse scattering method and its quantum counterpart, emphasizing the importance of integrable models in various fields such as condensed matter physics and string theory. The core of the lecture is the presentation of the XXX model, which is defined on a discrete circle with periodic boundary conditions. The model is described using a Lax operator, which is a key object in the quantum inverse scattering method. The lecture derives the Bethe Ansatz equations (BAE) for the XXX model, which are algebraic equations that determine the eigenstates and eigenvalues of the Hamiltonian. These equations are derived using the commutation relations of the Lax operator and the properties of the monodromy matrix. The lecture also discusses the thermodynamic limit of the XXX model, where the number of sites \( N \) approaches infinity. In this limit, the BAE simplify, and the spectrum of the model can be described in terms of quasiparticles with momentum and energy. The lecture explains how these quasiparticles interact through scattering processes, and how the Bethe Ansatz equations can be used to describe bound states and complex solutions. Finally, the lecture provides explicit expressions for the eigenvalues of important observables, such as the spin and momentum, on Bethe vectors. It also discusses the physical spectrum of the ferromagnetic phase of the model, including the ground state and excited states, and the scattering properties of the quasiparticles. The lecture concludes with a discussion of the mathematical and physical features of the thermodynamic limit, highlighting the differences between even and odd values of \( N \).