To analyze integrable multiparticle systems in the thermodynamic limit, one typically confines the system to a large periodic box, ensuring that the Bethe wave function resides in the asymptotic regime of free motion. The resulting periodic boundary conditions give rise to a system of coupled nonlinear equations for particle momenta, known as the Bethe equations. Solving these equations requires constructing common eigenvectors for a family of commuting operators acting on the Hilbert space of an auxiliary spin chain. This approach, known as the coordinate Bethe ansatz, is grounded in the generalized Bethe hypothesis, which posits the coordinate-space structure of eigenstates. This chapter introduces the formalism of the coordinate Bethe ansatz and its algebraic underpinnings, including the role of the Bethe–Yang equation, permutation modules, and Specht modules. The chapter concludes by applying this framework to the Lieb–Liniger model for spin-1/2 particles, setting the stage for more advanced developments.