  The asteroid belt is the leftover of the original planetesimal population in
the inner solar system. However, currently the asteroids have orbits with all
possible values of eccentricities and inclinations compatible with long-term
dynamical stability, whereas the initial planetesimal orbits should have been
quasi-circular and almost co-planar. The total mass in the asteroid population
is a small fraction of that existing primordially. Also, asteroids with
different chemical/mineralogical properties are not ranked in an orderly manner
with mean heliocentric distance as one could expect from the existence of a
radial gradient of the temperature in the proto-planetary disk, but they are
partially mixed. These properties show that the asteroid belt has been severely
sculpted by one or a series of processes during its lifetime. This paper
reviews the processes that have been proposed so far, discussing the properties
that they explain and the problems that they are confronted with. Emphasis is
paid to the interplay between the dynamical and the collisional evolution of
the asteroid population, which allows the use of the size distribution to
constrain the dynamical models. We divide the asteroid belt evolution into
three phases. The first phase started during the lifetime of the gaseous
proto-planetary disk, when the giant planets formed and presumably experienced
large-scale migrations, and continued after the removal of the gas, during the
build-up of the terrestrial planets. The second phase occurred after the
removal of the gaseous proto-planetary disk and it became particularly lively
for the asteroid belt when the giant planets suddenly changed their orbits, as
a result of a mutual dynamical instability and the interaction with the
trans-Neptunian planetesimal disk. The third phase covers the aftermath of the
giant planet instability, until today.
