The goal of this paper is the numerical computation and continuation of homoclinic connections of the Lyapunov families of periodic orbits (p.o.) associated with the collinear equilibrium points, L1, L2 and L3, of the planar circular restricted three-body problem (RTBP). We describe the method used that allows us to follow individual families of homoclinic connections by numerical continuation of a system of (nonlinear) equations that has as unknowns the initial condition of the p.o., the linear approximation of its stable and unstable manifolds and a point in a given Poincaré section in which the unstable and stable manifolds match. For the L3 case, some comments are made on the geometry of the manifold tubes and the possibility of obtaining trajectories with prescribed itineraries. © 2009 IOP Publishing Ltd and London Mathematical Society.