Colloidal semiconductor quantum dots connected by organic or inorganic molecules can form periodic supercrystals. These supercrystals can be used for various types of electronic and optical applications with properties superior to those of random quantum dots and organic polymer mixtures. We have used ab initio calculations to study the charge transport and carrier mobility in such supercrystals. Among the different possible charge transport mechanisms, we found that the phonon-assisted hopping is the most likely mechanism. The calculated carrier mobility agrees well with the experimentally measured results. Our predictions of the size and temperature dependences on the mobility are awaiting experimental confirmation.