  Based on the observed high velocity of pulsars it is thought that neutron
stars (NSs) receive a significant velocity kick at birth. Such natal kicks are
considered to play an important role in the the evolution of binary-NS systems.
The kick given to the NS (together with the effect of mass loss due to the
supernova explosion of the NS progenitor) may result in the binary disruption
or lead to a significant change of the binary orbital properties. Here we
explore in detail the dynamical aftermath of natal kicks in binary systems,
determine their possible outcomes and characterize their relative frequency,
making use of analytic arguments and detailed population synthesis models. In a
fraction of the cases the kick may cast the NS in such a trajectory as to
collide with the binary companion, or pass sufficiently close to it as to
disrupt it (micro tidal disruption event; $\mu$TDE), or alternatively it could
be tidally-captured into a close orbit, eventually forming an X-ray binary. We
calculate the rates of direct post-kick physical collisions and the possible
potential production of Thorne-Zytkow objects or long-GRBs through this
process, estimate the rates X-ray binaries formation and determine the rates of
$\mu$TDEs and faint supernovae from white dwarf disruptions by NSs. Finally we
suggest that natal kicks can produce BH-NS binaries with very short
gravitational-wave merger time, possibly giving rise to a new type of promptly
appearing eLISA gravitational wave (GW) sources, as well as producing aLIGO
binary-merger GW sources with a unique (likely type Ib/c) supernova
electromagnetic counterpart which precedes the GW merger.
