  The orbital dynamics of a spacecraft, or a comet, or an asteroid in the
Earth-Moon system in a scattering region around the Moon using the three
dimensional version of the circular restricted three-body problem is
numerically investigated. The test particle can move in bounded orbits around
the Moon or escape through the openings around the Lagrange points $L_1$ and
$L_2$ or even collide with the surface of the Moon. We explore in detail the
first four of the five possible Hill's regions configurations depending on the
value of the Jacobi constant which is of course related with the total orbital
energy. We conduct a thorough numerical analysis on the phase space mixing by
classifying initial conditions of orbits in several two-dimensional types of
planes and distinguishing between four types of motion: (i) ordered bounded,
(ii) trapped chaotic, (iii) escaping and (iv) collisional. In particular, we
locate the different basins and we relate them with the corresponding spatial
distributions of the escape and collision times. Our outcomes reveal the high
complexity of this planetary system. Furthermore, the numerical analysis
suggests a strong dependence of the properties of the considered basins with
both the total orbital energy and the initial value of the $z$ coordinate, with
a remarkable presence of fractal basin boundaries along all the regimes. Our
results are compared with earlier ones regarding the planar version of the
Earth-Moon system.
