Calcium-silicate hydrate (C-S-H) is the main binding agent in cement and concrete. It forms at the beginning of cement hydration, it progressively densifies as cement hardens and is ultimately responsible of concrete performances. This hydration product is a cohesive nano-scale gel, whose structure and mechanics are still poorly understood, in spite of its practical importance. Here we review some of the open questions for this fascinating material and a statistical physics approach recently developed, which allows us to investigate the gel formation under the out-of-equilibrium conditions typical of cement hydration and the role of the nano-scale structure in C-S-H mechanics upon hardening. Our approach unveils how some distinctive features of the kinetics of cement hydration can be related to changes in the morphology of the gels and elucidates the role of nano-scale mechanical heterogeneities in the hardened C-S-H.