  This review is a kinetic theory study investigating the effects of
inelasticity on the structure of the non-equilibrium states, in particular on
the behavior of the velocity distribution in the high energy tails. Starting
point is the nonlinear Boltzmann equation for spatially homogeneous systems,
which supposedly describes the behavior of the velocity distribution function
in dissipative systems as long as the system remains in the homogeneous cooling
state, i.e. on relatively short time scales before the clustering and similar
instabilities start to create spatial inhomogeneities. This is done for the two
most common models for dissipative systems, i.e. inelastic hard spheres and
inelastic Maxwell particles. In systems of Maxwell particles the collision
frequency is independent of the relative velocity of the colliding particles,
and in hard sphere systems it is linear. We then demonstrate the existence of
scaling solutions for the velocity distribution function, $F(v,t) \sim
v_0(t)^{-d} f((v/v_0(t))$, where $v_0$ is the r.m.s. velocity. The scaling form
$f(c)$ shows overpopulation in the high energy tails. In the case of freely
cooling systems the tails are of algebraic form, $ f(c)\sim c^{-d-a}$, where
the exponent $a$ may or may not depend on the degree of inelasticity, and in
the case of forced systems the tails are of stretched Gaussian type
$f(v)\sim\exp[-\beta (v/v_0)^b]$ with $b <2$.
