We will discuss recent results on the rigidity of convex sets in hyperbolic 3-space and their connection with discrete uniformization of polyhedral surfaces. A central idea is to realize discrete conformal changes of polyhedral metrics through boundaries of convex sets in hyperbolic 3-space. This viewpoint leads to uniformization results for all polyhedral surfaces, including discrete analogues of classical uniformization and the Riemann mapping theorem. Specifically, we will discuss a rigidity theorem for noncompact convex sets in hyperbolic 3-space, extending Pogorelov-type rigidity under a natural condition on the limit set at infinity, and explain how this rigidity enters the discrete uniformization of polyhedral surfaces. The talk is based on joint work with Yanwen Luo, Zhenghao Rao, and Xinrong Zhao.