Stars with. initial masses in the range 0.89 less-than-or-equal-to M/M . less-than-or-equal-to 5.0 have been evolved from the main-sequence phase through to the end of the asymptotic giant branch (AGB). The calculations were done with metallicities Z of 0.016, 0.008, 0.004, and 0.001 to allow comparison with Galactic and Magellanic Cloud stars. The novel feature of these calculations is the inclusion of mass loss on the AGB using an empirical formula relating the mass-loss rate to the pulsation period. The calculations show that a superwind phase develops naturally on the AGB, but only over the last 2-3 thermal pulse cycles. Since the superwind operates only during the latter part of the quiescent (hydrogen burning) phases when the luminosity is high, most AGB stars probably experience several superwind phases between which the star would appear as a normal optically visible red giant. Estimates are made, as a function of mass and metallicity, for the lifetimes of the thermally pulsing AGB phase, the optically visible AGB phase, and the maximum pulsation periods for OH/IR and other pulsating dust-enshrouded AGB stars. The maximum AGB luminosities predicted from this work for stars with initial masses less than or similar to 3 M. are in excellent agreement with those observed for stars in Magellanic Cloud clusters. It is argued that current observational estimates of maximum AGB luminosities for more massive cluster stars are too faint. The initial-final mass relation should therefore be reliable, at least for initial masses less than or similar to 3 M., although it predicts white dwarf masses approximately 0.1 M., larger than the current observational calibration of the relation. In the stars of initial mass 5.0 M., helium shell flashes are found to be weak, and the evolutionary tracks are overluminous with respect to the classical AGB luminosity-core mass relation provided the envelope mass is large (greater than or similar to 1.5 M.). Such stars evolve down the AGB as envelope mass is reduced during the superwind phase. Finally, we note that the calculations fail to reproduce the high frequency of low-mass carbon stars which is observed in the Magellanic Clouds, a failure evident in other computations of AGB evolution.
