  The influence of extended convective mixing (overshoot) on asymptotic giant
branch stellar evolution is investigated in detail. The extended mixing is
treated time-dependently, and the efficiency declines exponentially with the
geometric distance from the convective boundary. It has been considered at all
convective boundaries, including the He-flash convection zone in the intershell
region which forms during the thermal pulses. Both the structural and the
chemical evolution are affected by the inclusion of overshoot. The main results
include a very efficient third dredge-up which leads to the formation of carbon
stars of low mass and luminosity. A C13 pocket which may serve as a neutron
source for the s-process can form after the third dredge-up has reached into
the C12 rich intershell. Overshoot applied to the pulse-driven convective zone
during the He-flash leads to a deeper penetration of the bottom of this
convective zone into the C/O core below the He-burning shell. This in turn
causes He4 to be less abundant in the intershell while C12 and O16 are more
abundant compared to calculations without overshoot. We show that overshoot at
the He-flash convection zone as well as at the base of the envelope convection
enhance the efficiency of the third dredge-up. Characteristic properties of the
structural and chemical evolution of AGB stars are presented.
