"Models of the Thermally Pulsing Asymptotic Giant Branch (TP-AGB) stellar evolutionary phase play a critical role across astrophysics, from the chemical composition of meteorites belonging to the pre-solar nebula up to galaxy evolution in the high-redshift Universe. In spite of its importance, the modelling of TP-AGB is still affected by large uncertainties that propagate into the field of extragalactic astronomy, degrading the predicting power of current population synthesis models of galaxies. The major goal of this proposal is to remedy this persistent condition of uncertainty and controversy. The solution to the TP-AGB star conundrum will be provided by a new approach, which stands on the optimised integration of a) state-of-the-art theoretical tools to account for the complex physics of TP-AGB stars (evolution, nucleosynthesis, pulsation, winds, dust formation, etc.), and b) exceptionally high-quality observations of resolved TP-AGB stellar populations in stars clusters and nearby galaxies (Magellanic Clouds, M31, dwarf galaxies up to 4 Mpc) with reliable measurements of their star formation histories. We will adopt a global calibration method, in which TP-AGB evolution models are required to simultaneously reproduce a set of well-defined observational constraints (distributions of luminosities, colours, pulsation periods, dust mass-loss rates, expansion velocities of dusty envelopes, etc.). This project will deepen our understanding of TP-AGB physics profoundly, and provide wide-spread community benefits as well. We will publicly release well-tested and reliable ``TP-AGB products'', including stellar tracks, isochrones in all photometric systems, and chemical yields for both gas and dust. Eventually these products will be embedded in the stellar population synthesis models that are routinely used to analyse the integrated galaxy observables that probe the extragalactic Universe."