  We consider deformations of the $SU(3)$ Affine Toda theory (AT) and
investigate the integrability properties of the deformed theories. We find that
for some special deformations all conserved quantities change to being
conserved only asymptotically, {\it i.e.} in the process of the scattering of
two solitons these charges do vary in time, but they return, after the
scattering, to the values they had prior to the scattering. This phenomenon,
which we have called quasi-integrability, is related to special properties of
the two-soliton solutions under space-time parity transformations. Some
properties of the AT solitons are discussed, especially those involving
interesting static multi-soliton solutions. We support our analytical studies
with detailed numerical ones in which the time evolution has been simulated by
the 4th order Runge-Kutta method. We find that for some perturbations the
solitons repel and for the others they form a quasi-bound state. When we send
solitons towards each other they can repel when they come close together with
or without `flipping' the fields of the model. The solitons radiate very little
and appear to be stable. These results support the ideas of
quasi-integrability, {\it i.e.} that many effects of integrability also
approximately hold for the deformed models.
