  The kinetic foundations of Tsallis' nonextensive thermostatistics are
investigated through Boltzmann's transport equation approach. Our analysis
follows from a nonextensive generalization of the ``molecular chaos
hypothesis". For $q>0$, the $q$-transport equation satisfies an $H$-theorem
based on Tsallis entropy. It is also proved that the collisional equilibrium is
given by Tsallis' $q$-nonextensive velocity distribution.
