Rather than providing a comprehensive survey on manifolds curved from below, the paper is aimed at. exhibiting and discussing some of the main ideas and tools that have been developed over decades. <...> For the same reason, only a relatively small sample of results is presented to illustrate this development and in doing this, simplicity is emphasized over generality. <...> In the same vein, at most a glimpse of an idea or strategy of a proof is given! <...>