The Three Body Problem (3BP) has been one of the main celestial mechanics problems in the past few centuries, for a long time studied by both engineers and mathematicians. Its practical purpose is to describe the motion of three particles only under their mutual gravitational interaction, thus it actually represents a ‘model’ of the real physical world. In fact, it can be seen as an extension of the Two Body Problem (2BP), which solutions are well-know and have been already employed in many space missions since the launch of Sputnik-1 in 1957, first artificial Earth satellite. Alternative formulation is given with the “restricted problem”, assuming the mass of the third body as negligible with respect to both principal ones. Under this assumption, justified by the small size of a general spacecraft compared to planets or moons, the ‘unperturbed’ motion of the main masses can be described by a Kepler solution, involving elliptic or circular bounded orbits. It follows the existence of an Elliptic (ER3BP) and a Circular problem (CR3BP), both admitting five equilibrium points (named Lagrange points), where a periodic orbital motion is theoretically possible. In this work three main families of periodic solutions (here called Lagrange orbits) have been investigated in a neighbourhood of L1/L2, Lagrange points adopted in the last 40 years for many space missions, e.g. for space observation and exploration purposes. The main objective here is to study these models based on their ‘standard’ formulation, so adopting the Dynamical System Theory for the Circular problem and later extending the entire discussion to the Elliptic one. In the CR3BP periodic solutions have been found embedded within continuous families, also showing different types of bifurcation. A single shooting method (Differential Correction algorithm) and a numerical continuation scheme have been applied, starting with the analytic approximation based on Perturbation Theory (Linstedt-Poincaré method). Indeed, the linear stability assessment, through Variational equations (studying the so-called Monodromy matrix), has provided large insights into dynamical proprieties of the problem. In some cases, close members within the same family have shown a very different behaviour, in the limit of this linear analysis, but still allowing to well-define principal bifurcations in their continuous parameters. The analysis and all methodologies presented have been tested on a nominal system, here the Earth-Moon-Spacecraft one, while their validity can be trivially extended to several other restricted problems. The ER3BP has been the second fundamental step of this work, where many additional aspects have been presented, e.g. the loss of continuous families. Nonetheless, non-trivial difficulties arise within the analytic approach, while many insights for the new dynamics can be provided by an analytic approximation of such motion. In support, the numerical approach has been able to tackle the problem, thus improving not only the analysis on linear stability, but most important revealing the so-called “eccentricity-bifurcation”. The latter is one of the most peculiar aspects related to the Elliptic problem, which now involves a new time-constraint (nominally the shooting-time), and leads to two branches of solutions (Left/Right family or Peri/Apo group) in agreement with most recent literature. A very different behaviour has been shown between these branches, while only resonance orbits actually survive within the ER3BP, once again highlighting the essentiality of adopting this more complete model for an accurate real space mission design.