The restricted circular three-body problem is studied. All global families of periodic orbits adjacent to the libration points are found. A scenario for the evolution of orbits in the family is given. Chains of global families are highlighted; the chain begins at the triangular libration point, contains global families for the triangular and all collinear libration points, and ends with a family whose orbits are pressed against the main bodies. The evolution of global families in the chain associated with changes in the energy of the system is described. Planar and spatial orbits are studied.