Using as reference test model the Planar Circular Restricted Three Body Problem, this paper explores its Lagrangian Coherent Structures, as well as its Hyperbolic Lagrangian Coherent Structures. The purpose is to identify stable and unstable manifolds acting as separatrices between orbits with different qualitative behaviour and, therefore, relevant to the dynamics of the problem. Particular attention is given to the manifolds associated to the collinear libration points and to the practical stability regions around the triangular equilibrium points