  In this paper we investigate the space and velocity distributions of old
neutron stars (aged 109 to 1010 yr) in our Galaxy. Galactic old Neutron Stars
(NSs) population fills a torus-like area extending to a few tens kiloparsecs
above the galactic plane. The initial velocity distribution of NSs is not well
known, in this work we adopt a three component initial distribution, as given
by the contribution of kick velocities, circular velocities and Maxwellian
velocities. For the spatial initial distribution we use a Gamma function. We
then use Monte Carlo simulations to follow the evolution of the NSs under the
influence of the Paczy{\P}nski Galactic gravitational potential. Our
calculations show that NS orbits have a very large Galactic radial expansion
and that their radial distribution peak is quite close to their progenitors'
one. We also study the NS vertical distribution and find that it can well be
described by a double exponential low. Finally, we investigate the correlation
of the vertical and radial distribution and study the radial dependence of
scale-heights.
