This chapter addresses the long-standing challenge of constructing the Bethe wave function in large but finite volumes via the diagonalization of the transfer matrix within the spin representation. While the coordinate Bethe ansatz provides effective tools for specific models, it becomes increasingly cumbersome in systems with internal degrees of freedom, such as the Lieb–Liniger model. The algebraic Bethe ansatz offers a more general and systematic framework, particularly well suited for handling nested structures. Spin chains serve as a natural setting for this formalism, offering both mathematical richness and physical relevance as models of interacting quantum spins. Starting from the Heisenberg spin chain, the chapter introduces the algebraic structure underpinning the method, including the R-matrix formulation, quantum group symmetries, and the construction of the transfer matrix. To connect the algebraic formalism with thermodynamic behavior, the chapter explores the string hypothesis, which organizes solutions to the Bethe equations into regular complex patterns. This leads naturally to the Bethe–Takahashi equations, which govern the thermodynamic limit of integrable spin chains. These tools enable a tractable analysis of excited states and physical observables, establishing a foundation for applying the algebraic Bethe ansatz to a broader class of quantum integrable systems.