Molecular hydrogen (H2) is a fundamental component of galaxies, being the most abundant element in molecular clouds, where stars form, and an important source of radiative cooling at low temperature. With the advent of the ALMA telescope, a large amount of data about the distribution of H2 in galaxies has become available. However, the large majority of numerical simulations on galactic and cosmological scales still lacks the ability to directly follow the formation and dissociation of H2, and must rely on pre-calibrated sub-grid models to compare the results with observations. I will present a new model to self-consistently track the evolution of H2, including gas and dust shielding, H2 self-shielding, star formation (SF), supernova feedback, and extragalactic and local stellar radiation. I will discuss the results of a suite of hydrodynamic simulations of an isolated gas-rich galaxy at z=3, showing that the model can naturally reproduce the observed correlation between SF and H2 surface densities, without assuming any a priori dependence of SF on the H2 abundance. I will also present a study of the kinematics and dynamics of molecular gas in high-redshift quasars (z=6), where we investigate whether a central accreting black hole (BH) can significantly affect the H2 distribution in the host galaxy and generate molecular outflows.