Starting from Barnette's classical lower bound theorem for simplicial polytopes (LBT), and its proof via framework rigidity by Kalai, I'll then discuss corresponding theorems and conjectures, as well as proof techniques, for sub-classes and super-classes of interest. For example, we'll consider centrally-symmetric simplicial polytopes, flag polytopes, cubical polytopes, polytopes with one non-simplex facet, general polytopes, and analogues for polyhedral spheres and manifolds.