Most of the physical processes driving the Thermally-Pulsing Asymptotic Giant Branch (TP-AGB) evolution are not yet fully understood and they need to be modelled with parametrised descriptions. The uncertainties of the models affect the interpretation of the spectrophotometric properties of galaxies up to high-redshift. In the framework of the ERC - STARKEY project, the aim of this Thesis is to constrain the uncertain parameters, i.e. third dredge-up and mass-loss, that still affect the TP-AGB models. To this purpose, I perform detailed simulations of AGB star populations in the Small Magellanic Cloud (SMC) based on robust measurements of the space-resolved star formation history as derived from the deep near-infrared photometry of the VISTA survey of the Magellanic Clouds. I compare the resulting synthetic catalogues with high-quality observations of resolved stellar populations in the infrared passbands of 2MASS and Spitzer. A large grid of TP-AGB evolutionary tracks is computed with several combinations of third dredge-up and mass-loss prescriptions. By requiring the models to reproduce the star counts and the luminosity functions of the observed Oxygen-, Carbon-rich and extreme-AGB stars, I put quantitative constraints on the efficiencies of the third dredge-up and mass-loss. The observed luminosity functions in all the available infrared photometric filters are successfully reproduced by two set of models, one with a relatively high mass-loss efficiency for Oxygen-rich stars and the second with a lower mass-loss efficiency and a lower efficiency of the third dredge-up for the more massive TP-AGB stars, i.e. initial masses larger than three solar masses. On the basis of the best-fitting model I present a complete characterisation of the AGB population in terms of stellar parameters, including the predicted mass-loss rates, initial masses, and Carbon-to-Oxygen ratio. I use the TP-AGB models calibrated in the SMC to model the population of Long Period Variables (LPVs) in the Large Magellanic Cloud as observed by Gaia. The remarkable agreement between models and observations allows us to guide the interpretation of a new observational diagram that is able to photometrically distinguish the evolutionary stages, the initial masses and the chemical type of these stars. In the context of the Large Synoptic Survey Telescope (LSST) science collaboration, I produce catalogues containing the synthetic photometry of the Magellanic Clouds in the Gaia and LSST filters. These catalogues, together with the all-sky simulations of the Milky Way will be made available to the community through the NOAO Data Lab to help defining the observing strategy of the LSST mini-surveys. In addition, I simulate samples of AGB stars in Local Group dwarf galaxies and find a general agreement with the data. However, to properly consider these objects in the TP-AGB models calibration, the simulations should be improved to take into account the crowding effects and the different areas used for the star formation histories derivation and the AGB stars identification. Finally, the products of this work, namely calibrated stellar isochrones and pulsation periods of LPVs, will be publicly available and ready to use for the interpretation of the data coming from present and future observing facilities. The calibrated TP-AGB models may be included in population synthesis models used to probe the integrated light of galaxies in the extragalactic Universe
