TITLE: The Lagrangian solutions AUTHOR: Angel Jorba Departament de Matematica Aplicada i Analisi Universitat de Barcelona Gran Via 585 08007 Barcelona, Spain. E-mail: angel@maia.ub.es ABSTRACT: This chapter focuses on the dynamics in a neighbourhood of the five equilibrium points of the Restricted Three-Body Problem. The first section is devoted to the discussion of the linear behaviour near the five points. Then, the motion in the vicinity of the collinear points is considered, discussing the effective computation of the center manifold as a tool to describe the nonlinear dynamics in an extended neigbourhood of these points. This technique is then applied to the Earth-Moon case, showing the existence of periodic and quasi-periodic motions, including the well-known Halo orbits. Next, the dynamics near the triangular points is discussed, showing how normal forms can be used to effectively describe the motion nearby. The Lyapunov stability is also considered, showing how the stability is proved in the planar case, and why it is not proved in the spatial case. This section also discusses how to bound the amount of diffusion that could be present in the spatial case. Finally, in the last section we focus on the effect of perturbations. More concretely, we mention the Elliptic Restricted Three-Body Problem, the Bicircular problem and similar models that contain periodic and quasi-periodic time-dependent perturbations.