In recent years we have become aware of the hazard posed by impacts of
asteroids and comets on our planet.  Although the annual probability
of such an impact is very small, the consequences could be so severe
that it is wise to study this threat and its possible countermeasures.

Work is being done in order to discover all the potential impactors
large enough to be of concern, and to study their motion to verify
that they are not going to fall on the Earth in the next
decades/centuries.  The discovery work is done by a number of small
telescopes in various continents (mostly in the Northern hemisphere),
while the dynamical studies are carried out in a very small number of
very specialized centres.

In this talk I review the current status of the discovery and
follow-up activities going on worldwide, and will describe some recent
advances in the theory of close encounters, between small bodies and
the Earth, that lead to feasible strategies to deflect an impactor.