We present new synthetic models of the TP-AGB evolution. They are computed for 7 values of initial metal content (Z from 0.0001 to 0.03) and for initial masses between 0.5 and 5.0 Msun, thus extending the low- and intermediate-mass tracks of Girardi et al. (2000) until the beginning of the post-AGB phase. The calculations are performed by means of a synthetic code that incorporates many recent improvements, among which we mention: (1) the use of detailed and revised analytical relations to describe the evolution of quiescent luminosity, inter-pulse period, third dredge-up, hot bottom burning, pulse cycle luminosity variations, etc.; (2) the use of variable molecular opacities -- i.e. opacities consistent with the changing photospheric chemical composition -- in the integration of a complete envelope model, instead of the standard choice of scaled-solar opacities; (3) the use of formalisms for the mass-loss rates derived from pulsating dust-driven wind models of C- and O-rich AGB stars; and (4) the switching of pulsation modes between the first overtone and the fundamental one along the evolution, which has consequences in terms of the history of mass loss. It follows that, in addition to the time evolution on the HR diagram, the new models predict in a consistent fashion also variations in surface chemical compositions, pulsation modes and periods, and mass-loss rates. The onset and efficiency of the third dredge-up process are calibrated in order to reproduce basic observables like the carbon star luminosity functions in the Magellanic Clouds, and TP-AGB lifetimes (star counts) in Magellanic Cloud clusters. Forthcoming papers will present the theoretical isochrones and chemical yields derived from these tracks, and additional tests performed with the aid of a complete population synthesis code.
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